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Собрать тест ЕГЭТест на тему Неопределенный интеграл
05 сентября 2025 11:02
Ниже представлен тест по теме Неопределенный интеграл для 11 класса. Тип вопросов: открытый. Всего 20 вопросов. Тест выдает ответы после каждого задания.
Важное правило: записывайте неопределенный интеграл с константой интегрирования C.
- Вычислите ∫ x^2 dx.
- Вычислите ∫ (3x^4 + 2x^2 − 5) dx.
- Вычислите ∫ (2x + 1) dx.
- Вычислите ∫ e^{2x} dx.
- Вычислите ∫ x e^{x} dx.
- Вычислите ∫ sin x dx.
- Вычислите ∫ cos x dx.
- Вычислите ∫ cos(3x) dx.
- Вычислите ∫ sec^2 x dx.
- Вычислите ∫ tan x dx.
- Вычислите ∫ dx/(x^2 + 1).
- Вычислите ∫ x^{−1/2} dx.
- Вычислите ∫ x/(x^2 + 1) dx.
- Вычислите ∫ (1/x) dx.
- Вычислите ∫ (2x + 1)√(x + 1) dx.
- Вычислите ∫ (2x + 3) e^{x} dx.
- Вычислите ∫ x^2 e^{3x} dx.
- Вычислите ∫ dx/((x + 2)^2 + 1).
- Вычислите ∫ dx/(x^2 − 1).
- Вычислите ∫ dx/((x + 1)^2 + 1).
Ответы
- ∫ x^2 dx = (1/3) x^3 + C.
- ∫ (3x^4 + 2x^2 − 5) dx = (3/5) x^5 + (2/3) x^3 − 5x + C.
- ∫ (2x + 1) dx = x^2 + x + C.
- ∫ e^{2x} dx = (1/2) e^{2x} + C.
- ∫ x e^{x} dx = (x − 1) e^{x} + C.
- ∫ sin x dx = −cos x + C.
- ∫ cos x dx = sin x + C.
- ∫ cos(3x) dx = (1/3) sin(3x) + C.
- ∫ sec^2 x dx = tan x + C.
- ∫ tan x dx = −ln|cos x| + C (или ln|sec x| + C).
- ∫ dx/(x^2 + 1) = arctan x + C.
- ∫ x^{−1/2} dx = 2√x + C.
- ∫ x/(x^2 + 1) dx = (1/2) ln(x^2 + 1) + C.
- ∫ (1/x) dx = ln|x| + C.
- ∫ (2x + 1)√(x + 1) dx = (4/5)(x + 1)^{5/2} − (2/3)(x + 1)^{3/2} + C.
- ∫ (2x + 3) e^{x} dx = (2x + 1) e^{x} + C.
- ∫ x^2 e^{3x} dx = e^{3x} (9x^2 − 6x + 2)/27 + C.
- ∫ dx/((x + 2)^2 + 1) = arctan(x + 2) + C.
- ∫ dx/(x^2 − 1) = (1/2) ln| (x − 1)/(x + 1) | + C.
- ∫ dx/((x + 1)^2 + 1) = arctan(x + 1) + C.
Если нужно, могу адаптировать уровень сложности (чуть проще или чуть сложнее), добавить подсказки по методам решения (подстановка, интегрирование по частям, частичные дроби) или оформить тест в формате для печати.
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