Таблица производных и первообразных

Производная Функция Первообразная
\( f'(x) \) \( f(x) \) \( F(x) \)
1.\( 0 \)\( C \)\( Cx \)
2.\( 1 \)\( x \)\( \frac{x^2}{2} \)
3.\( n x^{n-1} \)\( x^n \)\( \frac{x^{n+1}}{n+1}, \; n \neq -1 \)
4.\( \frac{1}{2\sqrt{x}} \)\( \sqrt{x} \)\( * \)
5.\( -\frac{1}{x^2} \)\( \frac{1}{x} \)\( \ln x \)
6.\( e^x \)\( e^x \)\( e^x \)
7.\( a^x \ln a \)\( a^x \)\( \frac{a^x}{\ln a} \)
8.\( \frac{1}{x} \)\( \ln x \)\( * \)
9.\( \frac{1}{x \ln a} \)\( \log_a x \)\( * \)
10.\( \cos x \)\( \sin x \)\( -\cos x \)
11.\( -\sin x \)\( \cos x \)\( \sin x \)
12.\( \frac{1}{\cos^2 x} \)\( \operatorname{tg} x \)\( * \)
13.\( * \)\( \frac{1}{\cos^2 x} \)\( \operatorname{tg} x \)
14.\( -\frac{1}{\sin^2 x} \)\( \operatorname{ctg} x \)\( * \)
15.\( * \)\( \frac{1}{\sin^2 x} \)\( -\operatorname{ctg} x \)
16.\( \frac{1}{\sqrt{1 - x^2}} \)\( \arcsin x \)\( * \)
17.\( * \)\( \frac{1}{\sqrt{1 - x^2}} \)\( \arcsin x \)
18.\( -\frac{1}{\sqrt{1 - x^2}} \)\( \arccos x \)\( * \)
19.\( \frac{1}{1 + x^2} \)\( \operatorname{arctg} x \)\( * \)
20.\( * \)\( \frac{1}{1 + x^2} \)\( \operatorname{arctg} x \)
21.\( -\frac{1}{1 + x^2} \)\( \operatorname{arcctg} x \)\( * \)