Таблица интегралов элементарных функций

\int 0\cdot dx=C
\int 1\cdot dx=x+C
\int x^n dx=\frac{x^{n+1}}{n+1}+C \ (n\not= -1)
\int e^x dx=e^x+C
\int \frac{1}{x}dx=ln|x|+C
\int\frac{1}{\sqrt{1-x^2}}dx=\left\{\begin{array}{l}arcsin x+C\\-arccos x+C\end{array}\right.
\int\frac{1}{1+x^2}dx=\left\{\begin{array}{l}arctg x+C\\-arcctg x+C\end{array}\right.
\int a^xdx=\frac{a^x}{lna}+C
\int \sin x dx=-\cos x+C
\int \cos x dx=\sin x+C
\int \tg x dx=-ln|\cos x|+C
\int \ctg x dx=ln|\sin x| +C
\int\frac{1}{\cos^2x}dx=\tg x +C
\int\frac{1}{\sin^2 x}dx=-\ctg x +C
\int\frac{1}{\sin^2x}dx=-\ctg x +C
\int \frac{1}{\sin x}dx=ln\left|tg\frac{x}{2}\right|+C
\int\frac{1}{cos x}dx=ln\left|tg(\frac{x}{2}+\frac{\pi}{4})\right|+C
\int\frac{1}{\sqrt{a^2-x^2}}dx=\left\{\begin{array}{l}arcsin\frac{x}{a}+C\\-arccos\frac{x}{a}+C\end{array} \right.
\int\frac{1}{a^2+x^2}dx= \left\{ \begin{array}{l} \frac{1}{a}arctg\frac{x}{a}+C\\ -\frac{1} {a}arcctg\frac{x}{a}+C\end{array} \right.
\int\frac{1}{a^2-x^2}dx=\frac{1}{2a}ln|\frac{a+x}{a-x}|+C

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