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Номер 346, страница 184 - 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 7. Первообразная - номер 346, страница 184.
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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" => 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array:1 [ 0 => App\Models\Term {#1048 #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 {#1052 #items: array:2 [ 0 => App\Models\Term {#1053 #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 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+usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #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: [] 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"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 {#1060 #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 {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #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 [ 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"field_publication_number" => "17" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1073 #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: 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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 {#1075 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/kolmogorov" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:2 [ 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: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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Illuminate\Database\Eloquent\Collection {#1038 #items: array:6 [ 0 => App\Models\Element {#1104 #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" => 156408 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Edition {#1102 #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 {#1100 …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 {#1099 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 …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 {#1100 …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 {#1099 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1098 …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" => "104172" "type" => "task" ] "text" => "<p><strong>346.</strong> — Найдите общий вид первообразных для функции:</p><p><strong>a)</strong> $f(x) = 1 - \cos 3x + 2 \sin \left(\frac{\pi}{3} - x\right);$</p><p><strong>б)</strong> $f(x) = \frac{1}{\sin^2 4x} + \frac{1}{\sqrt{2-x}} - 3x^2;$</p><p><strong>в)</strong> $f(x) = \frac{2}{\cos^2 (3x+1)} - 3 \sin (4-x) + 2x;$</p><p><strong>г)</strong> $f(x) = \frac{1}{(3-2x)^3} + \frac{3}{\sqrt{5x-2}} - 2 \cos \left(\frac{\pi}{4} - x\right).$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "346-1.jpg" "alt" => null "width" => "1394" "height" => 529 "path" => "/media/algebra_10/kolmogorov/0-00/346-1.webp?ts=1733864924" ] ] ] #original: array:7 [ "id" => 156408 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1093} "task" => array:2 [ "refs" => "104172" "type" => "task" ] "text" => "<p><strong>346.</strong> — Найдите общий вид первообразных для функции:</p><p><strong>a)</strong> $f(x) = 1 - \cos 3x + 2 \sin \left(\frac{\pi}{3} - x\right);$</p><p><strong>б)</strong> $f(x) = \frac{1}{\sin^2 4x} + \frac{1}{\sqrt{2-x}} - 3x^2;$</p><p><strong>в)</strong> $f(x) = \frac{2}{\cos^2 (3x+1)} - 3 \sin (4-x) + 2x;$</p><p><strong>г)</strong> $f(x) = \frac{1}{(3-2x)^3} + \frac{3}{\sqrt{5x-2}} - 2 \cos \left(\frac{\pi}{4} - x\right).$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "346-1.jpg" "alt" => null "width" => "1394" "height" => 529 "path" => "/media/algebra_10/kolmogorov/0-00/346-1.webp?ts=1733864924" ] ] ] #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 {#1095 #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" => 157109 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Edition {#1096 #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 {#1092 …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 {#1088 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 …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 {#1092 …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 {#1088 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 …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" => "104172" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "346-1.png" "alt" => null "width" => "711" "height" => 1026 "path" => "/media/algebra_10/kolmogorov/1-00/346-1.webp?ts=1733865109" ] 1 => array:5 [ "name" => "346-2.png" "alt" => null "width" => "733" "height" => 1339 "path" => "/media/algebra_10/kolmogorov/1-00/346-2.webp?ts=1733865109" ] ] ] #original: array:6 [ "id" => 157109 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1082} "task" => array:2 [ "refs" => "104172" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "346-1.png" "alt" => null "width" => "711" "height" => 1026 "path" => "/media/algebra_10/kolmogorov/1-00/346-1.webp?ts=1733865109" ] 1 => array:5 [ "name" => "346-2.png" "alt" => null "width" => "733" "height" => 1339 "path" => "/media/algebra_10/kolmogorov/1-00/346-2.webp?ts=1733865109" ] ] ] #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 {#1086 #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" => 157772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Edition {#1087 #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 {#1084 …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 {#1043 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1034 …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 {#1084 …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 {#1043 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1034 …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" => "104172" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "346-1.jpg" "alt" => null "width" => "1113" "height" => 509 "path" => "/media/algebra_10/kolmogorov/2-00/346-1.webp?ts=1733995823" ] ] ] #original: array:6 [ "id" => 157772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113} "task" => array:2 [ "refs" => "104172" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "346-1.jpg" "alt" => null "width" => "1113" "height" => 509 "path" => "/media/algebra_10/kolmogorov/2-00/346-1.webp?ts=1733995823" ] ] ] #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 {#1132 #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" => 168499 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1142 #items: array:1 [ 0 => App\Models\Edition {#1133 #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 {#1136 …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 {#1137 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139 …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 {#1136 …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 {#1137 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139 …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" => "104172" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "346-1.jpg" "alt" => null "width" => "1113" "height" => 509 "path" => "/media/algebra_10/kolmogorov/3-00/346-1.webp?ts=1733995792" ] 1 => array:5 [ "name" => "346-2.jpg" "alt" => null "width" => "1113" "height" => 1435 "path" => "/media/algebra_10/kolmogorov/3-00/346-2.webp?ts=1733995792" ] ] ] #original: array:6 [ "id" => 168499 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1142} "task" => array:2 [ "refs" => "104172" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "346-1.jpg" "alt" => null "width" => "1113" "height" => 509 "path" => "/media/algebra_10/kolmogorov/3-00/346-1.webp?ts=1733995792" ] 1 => array:5 [ "name" => "346-2.jpg" "alt" => null "width" => "1113" "height" => 1435 "path" => "/media/algebra_10/kolmogorov/3-00/346-2.webp?ts=1733995792" ] ] ] #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 {#1039 #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" => 168000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1134 #items: array:1 [ 0 => App\Models\Edition {#1114 #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 {#1118 …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 {#1116 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1117 …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 {#1118 …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 {#1116 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1117 …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" => "104172" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "346-1.jpg" "alt" => null "width" => "1416" "height" => 993 "path" => "/media/algebra_10/kolmogorov/4-00/346-1.webp?ts=1733992611" ] ] ] #original: array:6 [ "id" => 168000 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1134} "task" => array:2 [ "refs" => "104172" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "346-1.jpg" "alt" => null "width" => "1416" "height" => 993 "path" => "/media/algebra_10/kolmogorov/4-00/346-1.webp?ts=1733992611" ] ] ] #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 {#1140 #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" => 1341791 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Edition {#1141 #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 {#1144 …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 {#1145 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1147 …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 {#1144 …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 {#1145 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1147 …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" => "104172" "type" => "task" ] "text" => "<p><strong>а)</strong> Для нахождения общего вида первообразных для функции $f(x) = 1 - \cos 3x + 2 \sin\left(\frac{\pi}{3} - x\right)$ необходимо найти ее неопределенный интеграл $F(x) = \int f(x) dx$.<br>Интегрируем функцию почленно, используя свойство линейности интеграла:<br>$F(x) = \int \left(1 - \cos 3x + 2 \sin\left(\frac{\pi}{3} - x\right)\right) dx = \int 1 dx - \int \cos 3x dx + 2 \int \sin\left(\frac{\pi}{3} - x\right) dx$.<br>Применим табличные интегралы и правило интегрирования для функции вида $g(kx+b)$, первообразная которой равна $\frac{1}{k}G(kx+b)$, где $G(x)$ — первообразная для $g(x)$.<br>1. Первообразная для константы 1: $\int 1 dx = x$.<br>2. Первообразная для $\cos 3x$: здесь $k=3$, $\int \cos(u)du = \sin(u)$, следовательно, $\int \cos 3x dx = \frac{1}{3}\sin 3x$.<br>3. Первообразная для $2 \sin\left(\frac{\pi}{3} - x\right)$: здесь $k=-1$, $\int \sin(u)du = -\cos(u)$, следовательно, $2 \int \sin\left(\frac{\pi}{3} - x\right) dx = 2 \cdot \frac{1}{-1} \left(-\cos\left(\frac{\pi}{3} - x\right)\right) = 2\cos\left(\frac{\pi}{3} - x\right)$.<br>Суммируя все найденные первообразные и добавляя произвольную постоянную $C$, получаем общий вид:<br>Ответ: $x - \frac{1}{3}\sin 3x + 2\cos\left(\frac{\pi}{3} - x\right) + C$.</p><p><strong>б)</strong> Для функции $f(x) = \frac{1}{\sin^2 4x} + \frac{1}{\sqrt{2-x}} - 3x^2$ найдем общий вид первообразных $F(x)$.<br>$F(x) = \int \left(\frac{1}{\sin^2 4x} + \frac{1}{\sqrt{2-x}} - 3x^2\right) dx = \int \frac{1}{\sin^2 4x} dx + \int (2-x)^{-1/2} dx - \int 3x^2 dx$.<br>1. Для $\int \frac{1}{\sin^2 4x} dx$: $k=4$, $\int \frac{1}{\sin^2 u} du = -\cot u$. Получаем $\frac{1}{4}(-\cot 4x) = -\frac{1}{4}\cot 4x$.<br>2. Для $\int (2-x)^{-1/2} dx$: $k=-1$, $\int u^{-1/2} du = 2u^{1/2} = 2\sqrt{u}$. Получаем $\frac{1}{-1}(2\sqrt{2-x}) = -2\sqrt{2-x}$.<br>3. Для $\int 3x^2 dx$: используем формулу для степенной функции, $\int x^n dx = \frac{x^{n+1}}{n+1}$. Получаем $3 \cdot \frac{x^3}{3} = x^3$.<br>Объединяем результаты и добавляем константу $C$:<br>Ответ: $-\frac{1}{4}\cot 4x - 2\sqrt{2-x} - x^3 + C$.</p><p><strong>в)</strong> Для функции $f(x) = \frac{2}{\cos^2(3x+1)} - 3 \sin(4-x) + 2x$ найдем общий вид первообразных $F(x)$.<br>$F(x) = \int \left(\frac{2}{\cos^2(3x+1)} - 3 \sin(4-x) + 2x\right) dx = 2\int \frac{1}{\cos^2(3x+1)} dx - 3\int \sin(4-x) dx + \int 2x dx$.<br>1. Для $2\int \frac{1}{\cos^2(3x+1)} dx$: $k=3$, $\int \frac{1}{\cos^2 u} du = \tan u$. Получаем $2 \cdot \frac{1}{3}\tan(3x+1) = \frac{2}{3}\tan(3x+1)$.<br>2. Для $-3\int \sin(4-x) dx$: $k=-1$, $\int \sin u du = -\cos u$. Получаем $-3 \cdot \frac{1}{-1}(-\cos(4-x)) = -3\cos(4-x)$.<br>3. Для $\int 2x dx$: получаем $2 \cdot \frac{x^2}{2} = x^2$.<br>Объединяем результаты и добавляем константу $C$:<br>Ответ: $\frac{2}{3}\tan(3x+1) - 3\cos(4-x) + x^2 + C$.</p><p><strong>г)</strong> Для функции $f(x) = \frac{1}{(3-2x)^3} + \frac{3}{\sqrt{5x-2}} - 2\cos\left(\frac{\pi}{4}-x\right)$ найдем общий вид первообразных $F(x)$.<br>$F(x) = \int \left((3-2x)^{-3} + 3(5x-2)^{-1/2} - 2\cos\left(\frac{\pi}{4}-x\right)\right) dx$.<br>Интегрируем каждое слагаемое отдельно:<br>1. Для $\int (3-2x)^{-3} dx$: $k=-2$, $\int u^{-3} du = \frac{u^{-2}}{-2}$. Получаем $\frac{1}{-2}\left(\frac{(3-2x)^{-2}}{-2}\right) = \frac{1}{4(3-2x)^2}$.<br>2. Для $3\int (5x-2)^{-1/2} dx$: $k=5$, $\int u^{-1/2} du = 2u^{1/2}$. Получаем $3 \cdot \frac{1}{5}(2\sqrt{5x-2}) = \frac{6}{5}\sqrt{5x-2}$.<br>3. Для $-2\int \cos\left(\frac{\pi}{4}-x\right) dx$: $k=-1$, $\int \cos u du = \sin u$. Получаем $-2 \cdot \frac{1}{-1}\sin\left(\frac{\pi}{4}-x\right) = 2\sin\left(\frac{\pi}{4}-x\right)$.<br>Объединяем результаты и добавляем константу $C$:<br>Ответ: $\frac{1}{4(3-2x)^2} + \frac{6}{5}\sqrt{5x-2} + 2\sin\left(\frac{\pi}{4}-x\right) + C$.</p>" ] #original: array:6 [ "id" => 1341791 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1150} "task" => array:2 [ "refs" => "104172" "type" => "task" ] "text" => "<p><strong>а)</strong> Для нахождения общего вида первообразных для функции $f(x) = 1 - \cos 3x + 2 \sin\left(\frac{\pi}{3} - x\right)$ необходимо найти ее неопределенный интеграл $F(x) = \int f(x) dx$.<br>Интегрируем функцию почленно, используя свойство линейности интеграла:<br>$F(x) = \int \left(1 - \cos 3x + 2 \sin\left(\frac{\pi}{3} - x\right)\right) dx = \int 1 dx - \int \cos 3x dx + 2 \int \sin\left(\frac{\pi}{3} - x\right) dx$.<br>Применим табличные интегралы и правило интегрирования для функции вида $g(kx+b)$, первообразная которой равна $\frac{1}{k}G(kx+b)$, где $G(x)$ — первообразная для $g(x)$.<br>1. Первообразная для константы 1: $\int 1 dx = x$.<br>2. Первообразная для $\cos 3x$: здесь $k=3$, $\int \cos(u)du = \sin(u)$, следовательно, $\int \cos 3x dx = \frac{1}{3}\sin 3x$.<br>3. Первообразная для $2 \sin\left(\frac{\pi}{3} - x\right)$: здесь $k=-1$, $\int \sin(u)du = -\cos(u)$, следовательно, $2 \int \sin\left(\frac{\pi}{3} - x\right) dx = 2 \cdot \frac{1}{-1} \left(-\cos\left(\frac{\pi}{3} - x\right)\right) = 2\cos\left(\frac{\pi}{3} - x\right)$.<br>Суммируя все найденные первообразные и добавляя произвольную постоянную $C$, получаем общий вид:<br>Ответ: $x - \frac{1}{3}\sin 3x + 2\cos\left(\frac{\pi}{3} - x\right) + C$.</p><p><strong>б)</strong> Для функции $f(x) = \frac{1}{\sin^2 4x} + \frac{1}{\sqrt{2-x}} - 3x^2$ найдем общий вид первообразных $F(x)$.<br>$F(x) = \int \left(\frac{1}{\sin^2 4x} + \frac{1}{\sqrt{2-x}} - 3x^2\right) dx = \int \frac{1}{\sin^2 4x} dx + \int (2-x)^{-1/2} dx - \int 3x^2 dx$.<br>1. Для $\int \frac{1}{\sin^2 4x} dx$: $k=4$, $\int \frac{1}{\sin^2 u} du = -\cot u$. Получаем $\frac{1}{4}(-\cot 4x) = -\frac{1}{4}\cot 4x$.<br>2. Для $\int (2-x)^{-1/2} dx$: $k=-1$, $\int u^{-1/2} du = 2u^{1/2} = 2\sqrt{u}$. Получаем $\frac{1}{-1}(2\sqrt{2-x}) = -2\sqrt{2-x}$.<br>3. Для $\int 3x^2 dx$: используем формулу для степенной функции, $\int x^n dx = \frac{x^{n+1}}{n+1}$. Получаем $3 \cdot \frac{x^3}{3} = x^3$.<br>Объединяем результаты и добавляем константу $C$:<br>Ответ: $-\frac{1}{4}\cot 4x - 2\sqrt{2-x} - x^3 + C$.</p><p><strong>в)</strong> Для функции $f(x) = \frac{2}{\cos^2(3x+1)} - 3 \sin(4-x) + 2x$ найдем общий вид первообразных $F(x)$.<br>$F(x) = \int \left(\frac{2}{\cos^2(3x+1)} - 3 \sin(4-x) + 2x\right) dx = 2\int \frac{1}{\cos^2(3x+1)} dx - 3\int \sin(4-x) dx + \int 2x dx$.<br>1. Для $2\int \frac{1}{\cos^2(3x+1)} dx$: $k=3$, $\int \frac{1}{\cos^2 u} du = \tan u$. Получаем $2 \cdot \frac{1}{3}\tan(3x+1) = \frac{2}{3}\tan(3x+1)$.<br>2. Для $-3\int \sin(4-x) dx$: $k=-1$, $\int \sin u du = -\cos u$. Получаем $-3 \cdot \frac{1}{-1}(-\cos(4-x)) = -3\cos(4-x)$.<br>3. Для $\int 2x dx$: получаем $2 \cdot \frac{x^2}{2} = x^2$.<br>Объединяем результаты и добавляем константу $C$:<br>Ответ: $\frac{2}{3}\tan(3x+1) - 3\cos(4-x) + x^2 + C$.</p><p><strong>г)</strong> Для функции $f(x) = \frac{1}{(3-2x)^3} + \frac{3}{\sqrt{5x-2}} - 2\cos\left(\frac{\pi}{4}-x\right)$ найдем общий вид первообразных $F(x)$.<br>$F(x) = \int \left((3-2x)^{-3} + 3(5x-2)^{-1/2} - 2\cos\left(\frac{\pi}{4}-x\right)\right) dx$.<br>Интегрируем каждое слагаемое отдельно:<br>1. Для $\int (3-2x)^{-3} dx$: $k=-2$, $\int u^{-3} du = \frac{u^{-2}}{-2}$. Получаем $\frac{1}{-2}\left(\frac{(3-2x)^{-2}}{-2}\right) = \frac{1}{4(3-2x)^2}$.<br>2. Для $3\int (5x-2)^{-1/2} dx$: $k=5$, $\int u^{-1/2} du = 2u^{1/2}$. Получаем $3 \cdot \frac{1}{5}(2\sqrt{5x-2}) = \frac{6}{5}\sqrt{5x-2}$.<br>3. Для $-2\int \cos\left(\frac{\pi}{4}-x\right) dx$: $k=-1$, $\int \cos u du = \sin u$. Получаем $-2 \cdot \frac{1}{-1}\sin\left(\frac{\pi}{4}-x\right) = 2\sin\left(\frac{\pi}{4}-x\right)$.<br>Объединяем результаты и добавляем константу $C$:<br>Ответ: $\frac{1}{4(3-2x)^2} + \frac{6}{5}\sqrt{5x-2} + 2\sin\left(\frac{\pi}{4}-x\right) + C$.</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" => "104173" "type" => "task" ] "previous" => array:2 [ "refs" => "104171" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\BookPage {#1110 #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" => 863543 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "184" "field_url" => "/10-klass/algebra/kolmogorov/page-184" "field_display_title" => "184" "field_folder" => "1" "field_image_name" => "184" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ 0 => App\Models\Book {#1049} ] #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 {#1111 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Element {#1156 #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" => 1037721 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1158 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1037721 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1158 …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" => "863544" "type" => "book_page" ] "previous" => array:2 [ "refs" => "863542" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:7 [ 0 => App\Models\Task {#1269 #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" => 104171 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "183" "field_page_end" => "184" "field_url" => "/10-klass/algebra/kolmogorov/345" "field_display_title" => "345" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1270 …2} "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 {#1271 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1272 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} …5 ] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Task {#1276 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] 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№346 (с. 184)
Условие. №346 (с. 184)
скриншот условия
346. — Найдите общий вид первообразных для функции:
a) $f(x) = 1 - \cos 3x + 2 \sin \left(\frac{\pi}{3} - x\right);$
б) $f(x) = \frac{1}{\sin^2 4x} + \frac{1}{\sqrt{2-x}} - 3x^2;$
в) $f(x) = \frac{2}{\cos^2 (3x+1)} - 3 \sin (4-x) + 2x;$
г) $f(x) = \frac{1}{(3-2x)^3} + \frac{3}{\sqrt{5x-2}} - 2 \cos \left(\frac{\pi}{4} - x\right).$
Решение 1. №346 (с. 184)
Решение 2. №346 (с. 184)
Решение 3. №346 (с. 184)
Решение 4. №346 (с. 184)
Решение 5. №346 (с. 184)
а) Для нахождения общего вида первообразных для функции $f(x) = 1 - \cos 3x + 2 \sin\left(\frac{\pi}{3} - x\right)$ необходимо найти ее неопределенный интеграл $F(x) = \int f(x) dx$.
Интегрируем функцию почленно, используя свойство линейности интеграла:
$F(x) = \int \left(1 - \cos 3x + 2 \sin\left(\frac{\pi}{3} - x\right)\right) dx = \int 1 dx - \int \cos 3x dx + 2 \int \sin\left(\frac{\pi}{3} - x\right) dx$.
Применим табличные интегралы и правило интегрирования для функции вида $g(kx+b)$, первообразная которой равна $\frac{1}{k}G(kx+b)$, где $G(x)$ — первообразная для $g(x)$.
1. Первообразная для константы 1: $\int 1 dx = x$.
2. Первообразная для $\cos 3x$: здесь $k=3$, $\int \cos(u)du = \sin(u)$, следовательно, $\int \cos 3x dx = \frac{1}{3}\sin 3x$.
3. Первообразная для $2 \sin\left(\frac{\pi}{3} - x\right)$: здесь $k=-1$, $\int \sin(u)du = -\cos(u)$, следовательно, $2 \int \sin\left(\frac{\pi}{3} - x\right) dx = 2 \cdot \frac{1}{-1} \left(-\cos\left(\frac{\pi}{3} - x\right)\right) = 2\cos\left(\frac{\pi}{3} - x\right)$.
Суммируя все найденные первообразные и добавляя произвольную постоянную $C$, получаем общий вид:
Ответ: $x - \frac{1}{3}\sin 3x + 2\cos\left(\frac{\pi}{3} - x\right) + C$.
б) Для функции $f(x) = \frac{1}{\sin^2 4x} + \frac{1}{\sqrt{2-x}} - 3x^2$ найдем общий вид первообразных $F(x)$.
$F(x) = \int \left(\frac{1}{\sin^2 4x} + \frac{1}{\sqrt{2-x}} - 3x^2\right) dx = \int \frac{1}{\sin^2 4x} dx + \int (2-x)^{-1/2} dx - \int 3x^2 dx$.
1. Для $\int \frac{1}{\sin^2 4x} dx$: $k=4$, $\int \frac{1}{\sin^2 u} du = -\cot u$. Получаем $\frac{1}{4}(-\cot 4x) = -\frac{1}{4}\cot 4x$.
2. Для $\int (2-x)^{-1/2} dx$: $k=-1$, $\int u^{-1/2} du = 2u^{1/2} = 2\sqrt{u}$. Получаем $\frac{1}{-1}(2\sqrt{2-x}) = -2\sqrt{2-x}$.
3. Для $\int 3x^2 dx$: используем формулу для степенной функции, $\int x^n dx = \frac{x^{n+1}}{n+1}$. Получаем $3 \cdot \frac{x^3}{3} = x^3$.
Объединяем результаты и добавляем константу $C$:
Ответ: $-\frac{1}{4}\cot 4x - 2\sqrt{2-x} - x^3 + C$.
в) Для функции $f(x) = \frac{2}{\cos^2(3x+1)} - 3 \sin(4-x) + 2x$ найдем общий вид первообразных $F(x)$.
$F(x) = \int \left(\frac{2}{\cos^2(3x+1)} - 3 \sin(4-x) + 2x\right) dx = 2\int \frac{1}{\cos^2(3x+1)} dx - 3\int \sin(4-x) dx + \int 2x dx$.
1. Для $2\int \frac{1}{\cos^2(3x+1)} dx$: $k=3$, $\int \frac{1}{\cos^2 u} du = \tan u$. Получаем $2 \cdot \frac{1}{3}\tan(3x+1) = \frac{2}{3}\tan(3x+1)$.
2. Для $-3\int \sin(4-x) dx$: $k=-1$, $\int \sin u du = -\cos u$. Получаем $-3 \cdot \frac{1}{-1}(-\cos(4-x)) = -3\cos(4-x)$.
3. Для $\int 2x dx$: получаем $2 \cdot \frac{x^2}{2} = x^2$.
Объединяем результаты и добавляем константу $C$:
Ответ: $\frac{2}{3}\tan(3x+1) - 3\cos(4-x) + x^2 + C$.
г) Для функции $f(x) = \frac{1}{(3-2x)^3} + \frac{3}{\sqrt{5x-2}} - 2\cos\left(\frac{\pi}{4}-x\right)$ найдем общий вид первообразных $F(x)$.
$F(x) = \int \left((3-2x)^{-3} + 3(5x-2)^{-1/2} - 2\cos\left(\frac{\pi}{4}-x\right)\right) dx$.
Интегрируем каждое слагаемое отдельно:
1. Для $\int (3-2x)^{-3} dx$: $k=-2$, $\int u^{-3} du = \frac{u^{-2}}{-2}$. Получаем $\frac{1}{-2}\left(\frac{(3-2x)^{-2}}{-2}\right) = \frac{1}{4(3-2x)^2}$.
2. Для $3\int (5x-2)^{-1/2} dx$: $k=5$, $\int u^{-1/2} du = 2u^{1/2}$. Получаем $3 \cdot \frac{1}{5}(2\sqrt{5x-2}) = \frac{6}{5}\sqrt{5x-2}$.
3. Для $-2\int \cos\left(\frac{\pi}{4}-x\right) dx$: $k=-1$, $\int \cos u du = \sin u$. Получаем $-2 \cdot \frac{1}{-1}\sin\left(\frac{\pi}{4}-x\right) = 2\sin\left(\frac{\pi}{4}-x\right)$.
Объединяем результаты и добавляем константу $C$:
Ответ: $\frac{1}{4(3-2x)^2} + \frac{6}{5}\sqrt{5x-2} + 2\sin\left(\frac{\pi}{4}-x\right) + C$.
Другие задания:
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