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Номер 354, страница 188 - гдз по алгебре 10-11 класс учебник Колмогоров, Абрамов
Авторы: Колмогоров А. Н., Абрамов А. М., Дудницын Ю. П.
Тип: Учебник
Издательство: Просвещение
Год издания: 2008 - 2026
Цвет обложки: зелёный, чёрный
ISBN: 978-5-09-019513-3
Рекомендовано Министерством образования и науки Российской Федерации
Алгебра и начала математического анализа
Популярные ГДЗ в 10 классе
Глава 3. Первообразная и интеграл. Параграф 8. Интеграл - номер 354, страница 188.
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null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 103772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "174" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1058 #items: array:1 [ 0 => App\Models\Book {#1057 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:1 [ 0 => App\Models\Term {#1059 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:2 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1063 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:3 [ 0 => App\Models\Term {#1079 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 4632 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Колмогоров" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Андрей" "field_patronymic" => "Николаевич" "field_surname_rp" => "Колмогорова" ] #original: array:12 [ "id" => 4632 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Колмогоров" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Андрей" "field_patronymic" => "Николаевич" "field_surname_rp" => "Колмогорова" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1077 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 3650 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абрамов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Михайлович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3650 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абрамов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Михайлович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1078 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 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"field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ 0 => App\Models\Term {#1071 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "17" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Term {#1082 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1084 #items: array:1 [ 0 => App\Models\Term {#1079} ] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10-11 классов общеобразовательных учреждений" "field_allowed" => "Рекомендовано Министерством образования и науки Российской Федерации" "field_reserve_field" => "Алгебра и начала математического анализа" "field_link_to_source" => null "field_tasks_count" => "1202" "field_priority" => "2" "field_default_folder" => "/algebra_10/kolmogorov/" "field_isbn" => "978-5-09-019513-3" "field_cover" => array:1 [ 0 => "/media/algebra_10/kolmogorov/covers/cover1.webp?ts=1733838489" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/kolmogorov/covers/cover1.webp?ts=1733838489" "alt" => "" "width" => "680" "height" => "915" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/kolmogorov" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:2 [ 0 => App\Models\Term {#1099 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 47 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"field_translit" => "chyornyy" ] #original: array:6 [ "id" => 56 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "чёрный" "field_cases" => array:6 [ …6] "field_translit" => "chyornyy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => "<!--Текст в низу страницы--> <p> Выбирая в качестве онлайн-консультанта <strong>ГДЗ по алгебре 10-11 класс учебник Колмогоров, Абрамов, Дудницын (Просвещение)</strong>, каждый школьник может быть уверен, что справится со всеми учебными задачами текущего года. Помимо идеальной подготовки домашних заданий, подросток сможет самостоятельно осваивать новые темы, выполнять диагностику своих знаний и восполнять в них пробелы, повторять и закреплять изученный материал. При этом ребенку не понадобятся дополнительные консультации педагога. Со всеми вопросами и темами текущего учебного года он справится самостоятельно. </p> <h2> Что готовит школьникам алгебра в 10-11 классах </h2> <p> Алгебру можно назвать лидером среди всех учебных дисциплин по количеству изучаемых тем. В десятом и одиннадцатом классах все усложняется тем, что помимо новых разделов, ребятам предстоит огромная работа по повторению ранее пройденного материала, ведь в конце года всех их ждет сложнейшее испытание - итоговый экзамен по предмету. Но его результат будет зависеть от того, насколько хорошо ребята освоят следующие разделы: </p> <ol> <li>Тригонометрические функции.</li> <li>Производная и её применения.</li> <li>Первообразная и интеграл.</li> <li>Показательная и логарифмическая функции.</li> <li>Действительные числа.</li> <li>Функции. </ol> <p> Материал очень насыщенный и достаточно сложный. При этом недостаточно будет просто вызубрить массу формул и правил. Простое заучивание - это лишь часть работы. Главное научиться применять полученные знания на практике. Именно с этим у большинства ребят возникают проблемы. Работа вместе с <strong>решебником по алгебре за 10-11 классы авторов Колмогорова, Абрамова, Дудницына</strong> позволит преодолеть все препятствия в учебе. Детально расписанная в пособии информация поможет прекрасно понять материал и научиться применять теорию на практике. </p> <h2> Что представляет собой решебник к учебнику Колмогорова (Просвещение) </h2> <p> Верные ответы — это далеко не все, что содержится во вспомогательном пособии. Сборник с готовыми домашними заданиями структурно и по содержанию полностью копирует учебник под редакцией Колмогорова и включает: </p> <ul> <li>тематические параграфы с соответствующими им упражнениями;</li> <li>номера к разделу «Задачи на повторение»;</li> <li>задачи повышенной трудности.</li> </ul> <p> Каждое решение расписано максимально подробно, с выкладками по теории, графиками и выводами. В особо сложных вопросах авторы дают дополнительные комментарии. Материал изложен простым языком, доступным для понимания абсолютно всем старшеклассникам. Электронный формат существенно облегчает работу с пособием, позволяет выбрать удобный ритм для учебы, экономит время и силы ребенка. </p> <h2> Зачем использовать ГДЗ по алгебре </h2> <p> Даже в выпускном классе нельзя опускать руки, если вдруг освоение алгебры не задалось. Еще есть время поправить свои оценки и знания. Главное правильно распределить свое время, найти верного помощника и вернуть интерес к предмету. Решебник позволит все эти элементы совместить воедино для достижения блестящего результата. Предложенное пособие станет тем консультантом, который даст возможность подростку быстро и качественно выполнить домашнее задание. Это в свою очередь позволит вернуть интерес к предмету и, что немало важно, выкроить ценные часы на отдых. Главное - не прибегать к списыванию готовых решений, ведь простое копирование поможет лишь на время, однако на первой же проверочной в классе обман вскроется. 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Помимо идеальной подготовки домашних заданий, подросток сможет самостоятельно осваивать новые темы, выполнять диагностику своих знаний и восполнять в них пробелы, повторять и закреплять изученный материал. При этом ребенку не понадобятся дополнительные консультации педагога. Со всеми вопросами и темами текущего учебного года он справится самостоятельно. </p> <h2> Что готовит школьникам алгебра в 10-11 классах </h2> <p> Алгебру можно назвать лидером среди всех учебных дисциплин по количеству изучаемых тем. В десятом и одиннадцатом классах все усложняется тем, что помимо новых разделов, ребятам предстоит огромная работа по повторению ранее пройденного материала, ведь в конце года всех их ждет сложнейшее испытание - итоговый экзамен по предмету. 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"185" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 103774 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Интеграл" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1134} "field_page_start" => "185" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1133} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:6 [ 0 => App\Models\Element {#1107 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 156423 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Edition {#1105 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 644 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "vadim" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_content_source" => null ] #original: array:21 [ "id" => 644 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "vadim" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "104180" "type" => "task" ] "text" => "<p><strong>354.</strong>-</p><p><strong>а)</strong> $y = x^3 + 1$, $y = 0$, $x = 0$, $x = 2$;</p><p><strong>б)</strong> $y = 1 + 2 \sin x$, $y = 0$, $x = 0$, $x = \frac{\pi}{2}$;</p><p><strong>в)</strong> $y = 4 - x^2$, $y = 0$;</p><p><strong>г)</strong> $y = 1 + \frac{1}{2} \cos x$, $y = 0$, $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "354-1.jpg" "alt" => null "width" => "1103" "height" => 332 "path" => "/media/algebra_10/kolmogorov/0-00/354-1.webp?ts=1733864929" ] ] ] #original: array:7 [ "id" => 156423 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098} "task" => array:2 [ "refs" => "104180" "type" => "task" ] "text" => "<p><strong>354.</strong>-</p><p><strong>а)</strong> $y = x^3 + 1$, $y = 0$, $x = 0$, $x = 2$;</p><p><strong>б)</strong> $y = 1 + 2 \sin x$, $y = 0$, $x = 0$, $x = \frac{\pi}{2}$;</p><p><strong>в)</strong> $y = 4 - x^2$, $y = 0$;</p><p><strong>г)</strong> $y = 1 + \frac{1}{2} \cos x$, $y = 0$, $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "354-1.jpg" "alt" => null "width" => "1103" "height" => 332 "path" => "/media/algebra_10/kolmogorov/0-00/354-1.webp?ts=1733864929" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1092 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 157138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Edition {#1095 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_content_source" => null ] #original: array:21 [ "id" => 645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "104180" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "354-1.png" "alt" => null "width" => "768" "height" => 1427 "path" => "/media/algebra_10/kolmogorov/1-00/354-1.webp?ts=1733865120" ] ] ] #original: array:6 [ "id" => 157138 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1051} "task" => array:2 [ "refs" => "104180" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "354-1.png" "alt" => null "width" => "768" "height" => 1427 "path" => "/media/algebra_10/kolmogorov/1-00/354-1.webp?ts=1733865120" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1052 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 157795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Edition {#1116 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_content_source" => null ] #original: array:21 [ "id" => 646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "104180" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "354-1.jpg" "alt" => null "width" => "1118" "height" => 366 "path" => "/media/algebra_10/kolmogorov/2-00/354-1.webp?ts=1733995833" ] ] ] #original: array:6 [ "id" => 157795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1125} "task" => array:2 [ "refs" => "104180" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "354-1.jpg" "alt" => null "width" => "1118" "height" => 366 "path" => "/media/algebra_10/kolmogorov/2-00/354-1.webp?ts=1733995833" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Element {#1122 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 168506 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1147 #items: array:1 [ 0 => App\Models\Edition {#1136 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1143 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_content_source" => null ] #original: array:21 [ "id" => 647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1143 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1144 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "104180" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "354-1.jpg" "alt" => null "width" => "1118" "height" => 366 "path" => "/media/algebra_10/kolmogorov/3-00/354-1.webp?ts=1733995801" ] 1 => array:5 [ "name" => "354-2.jpg" "alt" => null "width" => "1118" "height" => 773 "path" => "/media/algebra_10/kolmogorov/3-00/354-2.webp?ts=1733995801" ] ] ] #original: array:6 [ "id" => 168506 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1147} "task" => array:2 [ "refs" => "104180" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "354-1.jpg" "alt" => null "width" => "1118" "height" => 366 "path" => "/media/algebra_10/kolmogorov/3-00/354-1.webp?ts=1733995801" ] 1 => array:5 [ "name" => "354-2.jpg" "alt" => null "width" => "1118" "height" => 773 "path" => "/media/algebra_10/kolmogorov/3-00/354-2.webp?ts=1733995801" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Element {#1054 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 168008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:1 [ 0 => App\Models\Edition {#1049 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1034 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_content_source" => null ] #original: array:21 [ "id" => 648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1034 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "104180" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "354-1.jpg" "alt" => null "width" => "1416" "height" => 477 "path" => "/media/algebra_10/kolmogorov/4-00/354-1.webp?ts=1733992619" ] 1 => array:5 [ "name" => "354-2.jpg" "alt" => null "width" => "1416" "height" => 1472 "path" => "/media/algebra_10/kolmogorov/4-00/354-2.webp?ts=1733992619" ] ] ] #original: array:6 [ "id" => 168008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1139} "task" => array:2 [ "refs" => "104180" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "354-1.jpg" "alt" => null "width" => "1416" "height" => 477 "path" => "/media/algebra_10/kolmogorov/4-00/354-1.webp?ts=1733992619" ] 1 => array:5 [ "name" => "354-2.jpg" "alt" => null "width" => "1416" "height" => 1472 "path" => "/media/algebra_10/kolmogorov/4-00/354-2.webp?ts=1733992619" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1145 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1341819 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Edition {#1146 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5114 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1150 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5114 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1150 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "104180" "type" => "task" ] "text" => "<p><strong>а)</strong> Требуется найти площадь фигуры, ограниченной линиями $y = x^3 + 1$, $y = 0$ (ось Ox), $x = 0$ и $x = 2$. Такая фигура называется криволинейной трапецией.</p><p>На отрезке $[0, 2]$ функция $y = x^3 + 1$ принимает неотрицательные значения, так как при $x \ge 0$, $x^3 \ge 0$, и следовательно $x^3+1 \ge 1$. Площадь фигуры вычисляется с помощью определенного интеграла:</p><p>$S = \int_a^b f(x) \,dx = \int_0^2 (x^3 + 1) \,dx$</p><p>Найдем первообразную для подынтегральной функции $f(x) = x^3 + 1$. Первообразная равна $F(x) = \frac{x^4}{4} + x$.</p><p>Применим формулу Ньютона-Лейбница:</p><p>$S = \left[ \frac{x^4}{4} + x \right]_0^2 = \left(\frac{2^4}{4} + 2\right) - \left(\frac{0^4}{4} + 0\right) = \left(\frac{16}{4} + 2\right) - 0 = 4 + 2 = 6$.</p><p>Ответ: $6$</p><p><strong>б)</strong> Требуется найти площадь фигуры, ограниченной линиями $y = 1 + 2 \sin x$, $y = 0$, $x = 0$ и $x = \frac{\pi}{2}$.</p><p>На отрезке $[0, \frac{\pi}{2}]$ синус принимает значения от $0$ до $1$, поэтому функция $y = 1 + 2 \sin x$ является положительной ($1 \le 1 + 2 \sin x \le 3$). Площадь вычисляется как определенный интеграл:</p><p>$S = \int_0^{\pi/2} (1 + 2 \sin x) \,dx$</p><p>Первообразная для функции $f(x) = 1 + 2 \sin x$ равна $F(x) = x - 2 \cos x$.</p><p>Вычисляем интеграл по формуле Ньютона-Лейбница:</p><p>$S = \left[ x - 2 \cos x \right]_0^{\pi/2} = \left(\frac{\pi}{2} - 2 \cos\frac{\pi}{2}\right) - (0 - 2 \cos 0) = \left(\frac{\pi}{2} - 2 \cdot 0\right) - (0 - 2 \cdot 1) = \frac{\pi}{2} - (-2) = \frac{\pi}{2} + 2$.</p><p>Ответ: $\frac{\pi}{2} + 2$</p><p><strong>в)</strong> Требуется найти площадь фигуры, ограниченной линиями $y = 4 - x^2$ и $y = 0$.</p><p>Сначала найдем пределы интегрирования. Это точки пересечения графика функции с осью Ox. Для этого решим уравнение $4 - x^2 = 0$. Корни уравнения: $x_1 = -2$ и $x_2 = 2$. Это и будут наши пределы интегрирования, $a=-2$ и $b=2$.</p><p>На отрезке $[-2, 2]$ парабола $y = 4 - x^2$ находится выше оси Ox, то есть $4 - x^2 \ge 0$. Площадь вычисляется по формуле:</p><p>$S = \int_{-2}^2 (4 - x^2) \,dx$</p><p>Поскольку подынтегральная функция $f(x) = 4 - x^2$ является четной ($f(-x) = f(x)$), а пределы интегрирования симметричны относительно нуля, можно упростить вычисление:</p><p>$S = 2 \int_0^2 (4 - x^2) \,dx$</p><p>Первообразная для $f(x) = 4 - x^2$ есть $F(x) = 4x - \frac{x^3}{3}$.</p><p>$S = 2 \left[ 4x - \frac{x^3}{3} \right]_0^2 = 2 \left( \left(4 \cdot 2 - \frac{2^3}{3}\right) - \left(4 \cdot 0 - \frac{0^3}{3}\right) \right) = 2 \left( 8 - \frac{8}{3} \right) = 2 \left( \frac{24 - 8}{3} \right) = 2 \cdot \frac{16}{3} = \frac{32}{3}$.</p><p>Ответ: $\frac{32}{3}$</p><p><strong>г)</strong> Требуется найти площадь фигуры, ограниченной линиями $y = 1 + \frac{1}{2} \cos x$, $y = 0$, $x = -\frac{\pi}{2}$ и $x = \frac{\pi}{2}$.</p><p>На отрезке $[-\frac{\pi}{2}, \frac{\pi}{2}]$ косинус принимает значения от $0$ до $1$, поэтому функция $y = 1 + \frac{1}{2} \cos x$ положительна ($1 \le 1 + \frac{1}{2} \cos x \le 1.5$). Площадь вычисляется как определенный интеграл:</p><p>$S = \int_{-\pi/2}^{\pi/2} \left(1 + \frac{1}{2} \cos x\right) \,dx$</p><p>Подынтегральная функция $f(x) = 1 + \frac{1}{2} \cos x$ является четной, так как $\cos(-x) = \cos x$. Пределы интегрирования симметричны относительно нуля. Поэтому можно записать:</p><p>$S = 2 \int_0^{\pi/2} \left(1 + \frac{1}{2} \cos x\right) \,dx$</p><p>Первообразная для $f(x) = 1 + \frac{1}{2} \cos x$ равна $F(x) = x + \frac{1}{2} \sin x$.</p><p>$S = 2 \left[ x + \frac{1}{2} \sin x \right]_0^{\pi/2} = 2 \left( \left(\frac{\pi}{2} + \frac{1}{2} \sin\frac{\pi}{2}\right) - \left(0 + \frac{1}{2} \sin 0\right) \right) = 2 \left( \left(\frac{\pi}{2} + \frac{1}{2} \cdot 1\right) - 0 \right) = 2 \left( \frac{\pi}{2} + \frac{1}{2} \right) = \pi + 1$.</p><p>Ответ: $\pi + 1$</p>" ] #original: array:6 [ "id" => 1341819 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155} "task" => array:2 [ "refs" => "104180" "type" => "task" ] "text" => "<p><strong>а)</strong> Требуется найти площадь фигуры, ограниченной линиями $y = x^3 + 1$, $y = 0$ (ось Ox), $x = 0$ и $x = 2$. Такая фигура называется криволинейной трапецией.</p><p>На отрезке $[0, 2]$ функция $y = x^3 + 1$ принимает неотрицательные значения, так как при $x \ge 0$, $x^3 \ge 0$, и следовательно $x^3+1 \ge 1$. Площадь фигуры вычисляется с помощью определенного интеграла:</p><p>$S = \int_a^b f(x) \,dx = \int_0^2 (x^3 + 1) \,dx$</p><p>Найдем первообразную для подынтегральной функции $f(x) = x^3 + 1$. Первообразная равна $F(x) = \frac{x^4}{4} + x$.</p><p>Применим формулу Ньютона-Лейбница:</p><p>$S = \left[ \frac{x^4}{4} + x \right]_0^2 = \left(\frac{2^4}{4} + 2\right) - \left(\frac{0^4}{4} + 0\right) = \left(\frac{16}{4} + 2\right) - 0 = 4 + 2 = 6$.</p><p>Ответ: $6$</p><p><strong>б)</strong> Требуется найти площадь фигуры, ограниченной линиями $y = 1 + 2 \sin x$, $y = 0$, $x = 0$ и $x = \frac{\pi}{2}$.</p><p>На отрезке $[0, \frac{\pi}{2}]$ синус принимает значения от $0$ до $1$, поэтому функция $y = 1 + 2 \sin x$ является положительной ($1 \le 1 + 2 \sin x \le 3$). Площадь вычисляется как определенный интеграл:</p><p>$S = \int_0^{\pi/2} (1 + 2 \sin x) \,dx$</p><p>Первообразная для функции $f(x) = 1 + 2 \sin x$ равна $F(x) = x - 2 \cos x$.</p><p>Вычисляем интеграл по формуле Ньютона-Лейбница:</p><p>$S = \left[ x - 2 \cos x \right]_0^{\pi/2} = \left(\frac{\pi}{2} - 2 \cos\frac{\pi}{2}\right) - (0 - 2 \cos 0) = \left(\frac{\pi}{2} - 2 \cdot 0\right) - (0 - 2 \cdot 1) = \frac{\pi}{2} - (-2) = \frac{\pi}{2} + 2$.</p><p>Ответ: $\frac{\pi}{2} + 2$</p><p><strong>в)</strong> Требуется найти площадь фигуры, ограниченной линиями $y = 4 - x^2$ и $y = 0$.</p><p>Сначала найдем пределы интегрирования. Это точки пересечения графика функции с осью Ox. Для этого решим уравнение $4 - x^2 = 0$. Корни уравнения: $x_1 = -2$ и $x_2 = 2$. Это и будут наши пределы интегрирования, $a=-2$ и $b=2$.</p><p>На отрезке $[-2, 2]$ парабола $y = 4 - x^2$ находится выше оси Ox, то есть $4 - x^2 \ge 0$. Площадь вычисляется по формуле:</p><p>$S = \int_{-2}^2 (4 - x^2) \,dx$</p><p>Поскольку подынтегральная функция $f(x) = 4 - x^2$ является четной ($f(-x) = f(x)$), а пределы интегрирования симметричны относительно нуля, можно упростить вычисление:</p><p>$S = 2 \int_0^2 (4 - x^2) \,dx$</p><p>Первообразная для $f(x) = 4 - x^2$ есть $F(x) = 4x - \frac{x^3}{3}$.</p><p>$S = 2 \left[ 4x - \frac{x^3}{3} \right]_0^2 = 2 \left( \left(4 \cdot 2 - \frac{2^3}{3}\right) - \left(4 \cdot 0 - \frac{0^3}{3}\right) \right) = 2 \left( 8 - \frac{8}{3} \right) = 2 \left( \frac{24 - 8}{3} \right) = 2 \cdot \frac{16}{3} = \frac{32}{3}$.</p><p>Ответ: $\frac{32}{3}$</p><p><strong>г)</strong> Требуется найти площадь фигуры, ограниченной линиями $y = 1 + \frac{1}{2} \cos x$, $y = 0$, $x = -\frac{\pi}{2}$ и $x = \frac{\pi}{2}$.</p><p>На отрезке $[-\frac{\pi}{2}, \frac{\pi}{2}]$ косинус принимает значения от $0$ до $1$, поэтому функция $y = 1 + \frac{1}{2} \cos x$ положительна ($1 \le 1 + \frac{1}{2} \cos x \le 1.5$). Площадь вычисляется как определенный интеграл:</p><p>$S = \int_{-\pi/2}^{\pi/2} \left(1 + \frac{1}{2} \cos x\right) \,dx$</p><p>Подынтегральная функция $f(x) = 1 + \frac{1}{2} \cos x$ является четной, так как $\cos(-x) = \cos x$. Пределы интегрирования симметричны относительно нуля. Поэтому можно записать:</p><p>$S = 2 \int_0^{\pi/2} \left(1 + \frac{1}{2} \cos x\right) \,dx$</p><p>Первообразная для $f(x) = 1 + \frac{1}{2} \cos x$ равна $F(x) = x + \frac{1}{2} \sin x$.</p><p>$S = 2 \left[ x + \frac{1}{2} \sin x \right]_0^{\pi/2} = 2 \left( \left(\frac{\pi}{2} + \frac{1}{2} \sin\frac{\pi}{2}\right) - \left(0 + \frac{1}{2} \sin 0\right) \right) = 2 \left( \left(\frac{\pi}{2} + \frac{1}{2} \cdot 1\right) - 0 \right) = 2 \left( \frac{\pi}{2} + \frac{1}{2} \right) = \pi + 1$.</p><p>Ответ: $\pi + 1$</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "104181" "type" => "task" ] "previous" => array:2 [ "refs" => "104179" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\BookPage {#1113 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 863547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "188" "field_url" => "/10-klass/algebra/kolmogorov/page-188" "field_display_title" => "188" "field_folder" => "1" "field_image_name" => "188" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1114 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Element {#1161 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1037725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1163 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1037725 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1163 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "863548" "type" => "book_page" ] "previous" => array:2 [ "refs" => "863546" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1261 #items: array:4 [ 0 => App\Models\Task {#1271 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 104179 "created_at" => "2026-04-10 13:58:26" "updated_at" => null 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№354 (с. 188)
Условие. №354 (с. 188)
скриншот условия
354.-
а) $y = x^3 + 1$, $y = 0$, $x = 0$, $x = 2$;
б) $y = 1 + 2 \sin x$, $y = 0$, $x = 0$, $x = \frac{\pi}{2}$;
в) $y = 4 - x^2$, $y = 0$;
г) $y = 1 + \frac{1}{2} \cos x$, $y = 0$, $x = -\frac{\pi}{2}$, $x = \frac{\pi}{2}$.
Решение 1. №354 (с. 188)
Решение 2. №354 (с. 188)
Решение 3. №354 (с. 188)
Решение 4. №354 (с. 188)
Решение 5. №354 (с. 188)
а) Требуется найти площадь фигуры, ограниченной линиями $y = x^3 + 1$, $y = 0$ (ось Ox), $x = 0$ и $x = 2$. Такая фигура называется криволинейной трапецией.
На отрезке $[0, 2]$ функция $y = x^3 + 1$ принимает неотрицательные значения, так как при $x \ge 0$, $x^3 \ge 0$, и следовательно $x^3+1 \ge 1$. Площадь фигуры вычисляется с помощью определенного интеграла:
$S = \int_a^b f(x) \,dx = \int_0^2 (x^3 + 1) \,dx$
Найдем первообразную для подынтегральной функции $f(x) = x^3 + 1$. Первообразная равна $F(x) = \frac{x^4}{4} + x$.
Применим формулу Ньютона-Лейбница:
$S = \left[ \frac{x^4}{4} + x \right]_0^2 = \left(\frac{2^4}{4} + 2\right) - \left(\frac{0^4}{4} + 0\right) = \left(\frac{16}{4} + 2\right) - 0 = 4 + 2 = 6$.
Ответ: $6$
б) Требуется найти площадь фигуры, ограниченной линиями $y = 1 + 2 \sin x$, $y = 0$, $x = 0$ и $x = \frac{\pi}{2}$.
На отрезке $[0, \frac{\pi}{2}]$ синус принимает значения от $0$ до $1$, поэтому функция $y = 1 + 2 \sin x$ является положительной ($1 \le 1 + 2 \sin x \le 3$). Площадь вычисляется как определенный интеграл:
$S = \int_0^{\pi/2} (1 + 2 \sin x) \,dx$
Первообразная для функции $f(x) = 1 + 2 \sin x$ равна $F(x) = x - 2 \cos x$.
Вычисляем интеграл по формуле Ньютона-Лейбница:
$S = \left[ x - 2 \cos x \right]_0^{\pi/2} = \left(\frac{\pi}{2} - 2 \cos\frac{\pi}{2}\right) - (0 - 2 \cos 0) = \left(\frac{\pi}{2} - 2 \cdot 0\right) - (0 - 2 \cdot 1) = \frac{\pi}{2} - (-2) = \frac{\pi}{2} + 2$.
Ответ: $\frac{\pi}{2} + 2$
в) Требуется найти площадь фигуры, ограниченной линиями $y = 4 - x^2$ и $y = 0$.
Сначала найдем пределы интегрирования. Это точки пересечения графика функции с осью Ox. Для этого решим уравнение $4 - x^2 = 0$. Корни уравнения: $x_1 = -2$ и $x_2 = 2$. Это и будут наши пределы интегрирования, $a=-2$ и $b=2$.
На отрезке $[-2, 2]$ парабола $y = 4 - x^2$ находится выше оси Ox, то есть $4 - x^2 \ge 0$. Площадь вычисляется по формуле:
$S = \int_{-2}^2 (4 - x^2) \,dx$
Поскольку подынтегральная функция $f(x) = 4 - x^2$ является четной ($f(-x) = f(x)$), а пределы интегрирования симметричны относительно нуля, можно упростить вычисление:
$S = 2 \int_0^2 (4 - x^2) \,dx$
Первообразная для $f(x) = 4 - x^2$ есть $F(x) = 4x - \frac{x^3}{3}$.
$S = 2 \left[ 4x - \frac{x^3}{3} \right]_0^2 = 2 \left( \left(4 \cdot 2 - \frac{2^3}{3}\right) - \left(4 \cdot 0 - \frac{0^3}{3}\right) \right) = 2 \left( 8 - \frac{8}{3} \right) = 2 \left( \frac{24 - 8}{3} \right) = 2 \cdot \frac{16}{3} = \frac{32}{3}$.
Ответ: $\frac{32}{3}$
г) Требуется найти площадь фигуры, ограниченной линиями $y = 1 + \frac{1}{2} \cos x$, $y = 0$, $x = -\frac{\pi}{2}$ и $x = \frac{\pi}{2}$.
На отрезке $[-\frac{\pi}{2}, \frac{\pi}{2}]$ косинус принимает значения от $0$ до $1$, поэтому функция $y = 1 + \frac{1}{2} \cos x$ положительна ($1 \le 1 + \frac{1}{2} \cos x \le 1.5$). Площадь вычисляется как определенный интеграл:
$S = \int_{-\pi/2}^{\pi/2} \left(1 + \frac{1}{2} \cos x\right) \,dx$
Подынтегральная функция $f(x) = 1 + \frac{1}{2} \cos x$ является четной, так как $\cos(-x) = \cos x$. Пределы интегрирования симметричны относительно нуля. Поэтому можно записать:
$S = 2 \int_0^{\pi/2} \left(1 + \frac{1}{2} \cos x\right) \,dx$
Первообразная для $f(x) = 1 + \frac{1}{2} \cos x$ равна $F(x) = x + \frac{1}{2} \sin x$.
$S = 2 \left[ x + \frac{1}{2} \sin x \right]_0^{\pi/2} = 2 \left( \left(\frac{\pi}{2} + \frac{1}{2} \sin\frac{\pi}{2}\right) - \left(0 + \frac{1}{2} \sin 0\right) \right) = 2 \left( \left(\frac{\pi}{2} + \frac{1}{2} \cdot 1\right) - 0 \right) = 2 \left( \frac{\pi}{2} + \frac{1}{2} \right) = \pi + 1$.
Ответ: $\pi + 1$
Другие задания:
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