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Номер 3.2, страница 27 - гдз по алгебре 11 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
Популярные ГДЗ в 11 классе
Глава I. Первообразная и интеграл. Параграф 3. Определённый интеграл. Формула Ньютона — Лейбница - номер 3.2, страница 27.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #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" => 1124866 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "27" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/03-2" "field_display_title" => "3.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #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" => 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" => "№" ] #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" => 1124764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "I" "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 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "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 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "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" => "10" "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" => 4297 "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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "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 {#1064 #items: array:1 [ 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" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "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 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "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 {#1066 #items: array:1 [ 0 => 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" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #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 {#1060 #items: array:2 [ 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:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "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 => "*" ] } 1 => 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:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "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 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1068 #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" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #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_country" => Illuminate\Database\Eloquent\Collection {#1074 #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" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #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 {#1076 #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" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #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 {#1075 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Общественно-гуманитарного направления" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "530" "field_priority" => "2" "field_default_folder" => "/algebra_11/abylkasymova1-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 542 "class_subject" => 388 ] ] #original: array:50 [ "id" => 4297 "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} "field_class" => Illuminate\Database\Eloquent\Collection {#1064} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1066} "field_author" => Illuminate\Database\Eloquent\Collection {#1060} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1069} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1072} "field_country" => Illuminate\Database\Eloquent\Collection {#1074} "field_city" => Illuminate\Database\Eloquent\Collection {#1076} "field_series" => Illuminate\Database\Eloquent\Collection {#1075} "field_umk" => Illuminate\Database\Eloquent\Collection {#1070} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1077} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1078} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1079} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1080} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Общественно-гуманитарного направления" "field_allowed" => null "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "530" "field_priority" => "2" "field_default_folder" => "/algebra_11/abylkasymova1-u19/" "field_isbn" => null "field_cover" => array:1 [ 0 => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_11/abylkasymova1-u19/covers/cover1.webp?ts=1752223853" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1081} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1084} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 542 "class_subject" => 388 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "10" "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} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:2 [ 0 => App\Models\Branch {#1114 #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" => 1124764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "10" "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 {#1116 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Первообразная и интеграл" "field_branch_order" => "I" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1115} "field_page_start" => "10" "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 {#1116} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117} ] #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\Branch {#1118 #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" => 1124767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Определённый интеграл. Формула Ньютона — Лейбница" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Term {#1119 #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" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "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_page_start" => "24" "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 {#1121 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1124767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Определённый интеграл. Формула Ньютона — Лейбница" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1122} "field_page_start" => "24" "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 {#1121} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120} ] #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:2 [ 0 => App\Models\Element {#1097 #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" => 1308101 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1088 #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" => 4970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Term {#1063} ] #escapeWhenCastingToString: false } "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" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086} "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" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1051} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053} "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" => "1124866" "type" => "task" ] "text" => "<p><strong>3.2. 1)</strong> $\int_1^4 \frac{1}{\sqrt{x}} dx;$</p><p><strong>2)</strong> $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sin^2 x} dx;$</p><p><strong>3)</strong> $\int_0^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx;$</p><p><strong>4)</strong> $\int_9^{16} \frac{3}{\sqrt{x}} dx.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1307" "height" => 417 "path" => "/media/algebra_11/abylkasymova1-u19/0-03/2-1.webp?ts=1753819172" ] ] ] #original: array:7 [ "id" => 1308101 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1088} "task" => array:2 [ "refs" => "1124866" "type" => "task" ] "text" => "<p><strong>3.2. 1)</strong> $\int_1^4 \frac{1}{\sqrt{x}} dx;$</p><p><strong>2)</strong> $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sin^2 x} dx;$</p><p><strong>3)</strong> $\int_0^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx;$</p><p><strong>4)</strong> $\int_9^{16} \frac{3}{\sqrt{x}} dx.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1307" "height" => 417 "path" => "/media/algebra_11/abylkasymova1-u19/0-03/2-1.webp?ts=1753819172" ] ] ] #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" => 1553110 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1052 #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" => 5663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Term {#1043 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "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" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1041 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1105} "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" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1107} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1041} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "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" => "1124866" "type" => "task" ] "text" => "<p><strong>1)</strong> Для вычисления определенного интеграла $\int_{1}^{4} \frac{1}{\sqrt{x}} dx$ воспользуемся формулой Ньютона-Лейбница: $\int_{a}^{b} f(x) dx = F(b) - F(a)$, где $F(x)$ является первообразной для функции $f(x)$.</p><p>Сначала найдем первообразную для $f(x) = \frac{1}{\sqrt{x}}$. Для этого представим функцию в виде степени: $f(x) = x^{-1/2}$.</p><p>Первообразная для степенной функции $x^n$ находится по формуле $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.</p><p>Для $n = -1/2$, получаем: $F(x) = \frac{x^{-1/2 + 1}}{-1/2 + 1} = \frac{x^{1/2}}{1/2} = 2x^{1/2} = 2\sqrt{x}$.</p><p>Теперь подставляем в формулу Ньютона-Лейбница:</p><p>$\int_{1}^{4} \frac{1}{\sqrt{x}} dx = [2\sqrt{x}]_{1}^{4} = 2\sqrt{4} - 2\sqrt{1} = 2 \cdot 2 - 2 \cdot 1 = 4 - 2 = 2$.</p><p><strong>Ответ:</strong> $2$.</p><p><strong>2)</strong> Для вычисления интеграла $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sin^2 x} dx$ найдем первообразную для подынтегральной функции $f(x) = \frac{1}{\sin^2 x}$.</p><p>Это табличный интеграл, первообразная для которого равна $F(x) = -\cot x$.</p><p>Применим формулу Ньютона-Лейбница:</p><p>$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sin^2 x} dx = [-\cot x]_{\frac{\pi}{4}}^{\frac{\pi}{3}} = (-\cot(\frac{\pi}{3})) - (-\cot(\frac{\pi}{4})) = -\cot(\frac{\pi}{3}) + \cot(\frac{\pi}{4})$.</p><p>Зная значения котангенса: $\cot(\frac{\pi}{3}) = \frac{\sqrt{3}}{3}$ и $\cot(\frac{\pi}{4}) = 1$.</p><p>Подставляем и вычисляем: $-\frac{\sqrt{3}}{3} + 1 = 1 - \frac{\sqrt{3}}{3}$.</p><p><strong>Ответ:</strong> $1 - \frac{\sqrt{3}}{3}$.</p><p><strong>3)</strong> Вычислим интеграл $\int_{0}^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx$.</p><p>Первообразная для подынтегральной функции $f(x) = \frac{1}{\cos^2 x}$ является табличной: $F(x) = \tan x$.</p><p>Применим формулу Ньютона-Лейбница:</p><p>$\int_{0}^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx = [\tan x]_{0}^{\frac{\pi}{4}} = \tan(\frac{\pi}{4}) - \tan(0)$.</p><p>Значения тангенса: $\tan(\frac{\pi}{4}) = 1$ и $\tan(0) = 0$.</p><p>Подставляем и вычисляем: $1 - 0 = 1$.</p><p><strong>Ответ:</strong> $1$.</p><p><strong>4)</strong> Вычислим интеграл $\int_{9}^{16} \frac{3}{\sqrt{x}} dx$.</p><p>Вынесем постоянный множитель 3 за знак интеграла: $3 \int_{9}^{16} \frac{1}{\sqrt{x}} dx$.</p><p>Первообразная для функции $f(x) = \frac{1}{\sqrt{x}} = x^{-1/2}$ равна $F(x) = 2\sqrt{x}$ (как в пункте 1).</p><p>Применим формулу Ньютона-Лейбница:</p><p>$3 \int_{9}^{16} \frac{1}{\sqrt{x}} dx = 3 \cdot [2\sqrt{x}]_{9}^{16} = 3 \cdot (2\sqrt{16} - 2\sqrt{9})$.</p><p>Вычисляем значения корней: $\sqrt{16} = 4$ и $\sqrt{9} = 3$.</p><p>Подставляем и вычисляем: $3 \cdot (2 \cdot 4 - 2 \cdot 3) = 3 \cdot (8 - 6) = 3 \cdot 2 = 6$.</p><p><strong>Ответ:</strong> $6$.</p>" ] #original: array:6 [ "id" => 1553110 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1052} "task" => array:2 [ "refs" => "1124866" "type" => "task" ] "text" => "<p><strong>1)</strong> Для вычисления определенного интеграла $\int_{1}^{4} \frac{1}{\sqrt{x}} dx$ воспользуемся формулой Ньютона-Лейбница: $\int_{a}^{b} f(x) dx = F(b) - F(a)$, где $F(x)$ является первообразной для функции $f(x)$.</p><p>Сначала найдем первообразную для $f(x) = \frac{1}{\sqrt{x}}$. Для этого представим функцию в виде степени: $f(x) = x^{-1/2}$.</p><p>Первообразная для степенной функции $x^n$ находится по формуле $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.</p><p>Для $n = -1/2$, получаем: $F(x) = \frac{x^{-1/2 + 1}}{-1/2 + 1} = \frac{x^{1/2}}{1/2} = 2x^{1/2} = 2\sqrt{x}$.</p><p>Теперь подставляем в формулу Ньютона-Лейбница:</p><p>$\int_{1}^{4} \frac{1}{\sqrt{x}} dx = [2\sqrt{x}]_{1}^{4} = 2\sqrt{4} - 2\sqrt{1} = 2 \cdot 2 - 2 \cdot 1 = 4 - 2 = 2$.</p><p><strong>Ответ:</strong> $2$.</p><p><strong>2)</strong> Для вычисления интеграла $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sin^2 x} dx$ найдем первообразную для подынтегральной функции $f(x) = \frac{1}{\sin^2 x}$.</p><p>Это табличный интеграл, первообразная для которого равна $F(x) = -\cot x$.</p><p>Применим формулу Ньютона-Лейбница:</p><p>$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sin^2 x} dx = [-\cot x]_{\frac{\pi}{4}}^{\frac{\pi}{3}} = (-\cot(\frac{\pi}{3})) - (-\cot(\frac{\pi}{4})) = -\cot(\frac{\pi}{3}) + \cot(\frac{\pi}{4})$.</p><p>Зная значения котангенса: $\cot(\frac{\pi}{3}) = \frac{\sqrt{3}}{3}$ и $\cot(\frac{\pi}{4}) = 1$.</p><p>Подставляем и вычисляем: $-\frac{\sqrt{3}}{3} + 1 = 1 - \frac{\sqrt{3}}{3}$.</p><p><strong>Ответ:</strong> $1 - \frac{\sqrt{3}}{3}$.</p><p><strong>3)</strong> Вычислим интеграл $\int_{0}^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx$.</p><p>Первообразная для подынтегральной функции $f(x) = \frac{1}{\cos^2 x}$ является табличной: $F(x) = \tan x$.</p><p>Применим формулу Ньютона-Лейбница:</p><p>$\int_{0}^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx = [\tan x]_{0}^{\frac{\pi}{4}} = \tan(\frac{\pi}{4}) - \tan(0)$.</p><p>Значения тангенса: $\tan(\frac{\pi}{4}) = 1$ и $\tan(0) = 0$.</p><p>Подставляем и вычисляем: $1 - 0 = 1$.</p><p><strong>Ответ:</strong> $1$.</p><p><strong>4)</strong> Вычислим интеграл $\int_{9}^{16} \frac{3}{\sqrt{x}} dx$.</p><p>Вынесем постоянный множитель 3 за знак интеграла: $3 \int_{9}^{16} \frac{1}{\sqrt{x}} dx$.</p><p>Первообразная для функции $f(x) = \frac{1}{\sqrt{x}} = x^{-1/2}$ равна $F(x) = 2\sqrt{x}$ (как в пункте 1).</p><p>Применим формулу Ньютона-Лейбница:</p><p>$3 \int_{9}^{16} \frac{1}{\sqrt{x}} dx = 3 \cdot [2\sqrt{x}]_{9}^{16} = 3 \cdot (2\sqrt{16} - 2\sqrt{9})$.</p><p>Вычисляем значения корней: $\sqrt{16} = 4$ и $\sqrt{9} = 3$.</p><p>Подставляем и вычисляем: $3 \cdot (2 \cdot 4 - 2 \cdot 3) = 3 \cdot (8 - 6) = 3 \cdot 2 = 6$.</p><p><strong>Ответ:</strong> $6$.</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" => "1124867" "type" => "task" ] "previous" => array:2 [ "refs" => "1124865" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Book {#1057} ] 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#items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Branch {#1114} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1249 #items: array:2 [ 0 => App\Models\Branch {#1114} 1 => App\Models\Branch {#1118} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:2 [ 0 => App\Models\Element {#1268 …30} 1 => App\Models\Element {#1270 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1124871" "type" => "task" ] "previous" => array:2 [ "refs" => "1124869" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1265 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1100005" "type" => "book_page" ] ] #original: array:24 [ "id" => 1124870 "created_at" => "2026-04-10 13:58:26" "updated_at" => 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#classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Task {#1260 #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" => 1124871 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "27" "field_page_end" => null "field_url" => "/11-klass/algebra/mektep-abylkasymova1-uchebnik-2019/03-7" "field_display_title" => "3.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1252 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null 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#touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№3.2 (с. 27)
Условие. №3.2 (с. 27)
скриншот условия
3.2. 1) $\int_1^4 \frac{1}{\sqrt{x}} dx;$
2) $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sin^2 x} dx;$
3) $\int_0^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx;$
4) $\int_9^{16} \frac{3}{\sqrt{x}} dx.$
Решение 2 (rus). №3.2 (с. 27)
1) Для вычисления определенного интеграла $\int_{1}^{4} \frac{1}{\sqrt{x}} dx$ воспользуемся формулой Ньютона-Лейбница: $\int_{a}^{b} f(x) dx = F(b) - F(a)$, где $F(x)$ является первообразной для функции $f(x)$.
Сначала найдем первообразную для $f(x) = \frac{1}{\sqrt{x}}$. Для этого представим функцию в виде степени: $f(x) = x^{-1/2}$.
Первообразная для степенной функции $x^n$ находится по формуле $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.
Для $n = -1/2$, получаем: $F(x) = \frac{x^{-1/2 + 1}}{-1/2 + 1} = \frac{x^{1/2}}{1/2} = 2x^{1/2} = 2\sqrt{x}$.
Теперь подставляем в формулу Ньютона-Лейбница:
$\int_{1}^{4} \frac{1}{\sqrt{x}} dx = [2\sqrt{x}]_{1}^{4} = 2\sqrt{4} - 2\sqrt{1} = 2 \cdot 2 - 2 \cdot 1 = 4 - 2 = 2$.
Ответ: $2$.
2) Для вычисления интеграла $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sin^2 x} dx$ найдем первообразную для подынтегральной функции $f(x) = \frac{1}{\sin^2 x}$.
Это табличный интеграл, первообразная для которого равна $F(x) = -\cot x$.
Применим формулу Ньютона-Лейбница:
$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{1}{\sin^2 x} dx = [-\cot x]_{\frac{\pi}{4}}^{\frac{\pi}{3}} = (-\cot(\frac{\pi}{3})) - (-\cot(\frac{\pi}{4})) = -\cot(\frac{\pi}{3}) + \cot(\frac{\pi}{4})$.
Зная значения котангенса: $\cot(\frac{\pi}{3}) = \frac{\sqrt{3}}{3}$ и $\cot(\frac{\pi}{4}) = 1$.
Подставляем и вычисляем: $-\frac{\sqrt{3}}{3} + 1 = 1 - \frac{\sqrt{3}}{3}$.
Ответ: $1 - \frac{\sqrt{3}}{3}$.
3) Вычислим интеграл $\int_{0}^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx$.
Первообразная для подынтегральной функции $f(x) = \frac{1}{\cos^2 x}$ является табличной: $F(x) = \tan x$.
Применим формулу Ньютона-Лейбница:
$\int_{0}^{\frac{\pi}{4}} \frac{1}{\cos^2 x} dx = [\tan x]_{0}^{\frac{\pi}{4}} = \tan(\frac{\pi}{4}) - \tan(0)$.
Значения тангенса: $\tan(\frac{\pi}{4}) = 1$ и $\tan(0) = 0$.
Подставляем и вычисляем: $1 - 0 = 1$.
Ответ: $1$.
4) Вычислим интеграл $\int_{9}^{16} \frac{3}{\sqrt{x}} dx$.
Вынесем постоянный множитель 3 за знак интеграла: $3 \int_{9}^{16} \frac{1}{\sqrt{x}} dx$.
Первообразная для функции $f(x) = \frac{1}{\sqrt{x}} = x^{-1/2}$ равна $F(x) = 2\sqrt{x}$ (как в пункте 1).
Применим формулу Ньютона-Лейбница:
$3 \int_{9}^{16} \frac{1}{\sqrt{x}} dx = 3 \cdot [2\sqrt{x}]_{9}^{16} = 3 \cdot (2\sqrt{16} - 2\sqrt{9})$.
Вычисляем значения корней: $\sqrt{16} = 4$ и $\sqrt{9} = 3$.
Подставляем и вычисляем: $3 \cdot (2 \cdot 4 - 2 \cdot 3) = 3 \cdot (8 - 6) = 3 \cdot 2 = 6$.
Ответ: $6$.
Другие задания:
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