基本积分表

常数项：

\[ \int k \, dx=kx+c \]

\[ \begin{aligned} \int x^{\mu } \, dx=\frac{x^{\mu +1}}{\mu +1}(\mu \neq -1) \newline \int \frac{1}{x} \, dx=\ln \vert x \vert +c \end{aligned} \]

1+-平方函数：

\[ \begin{aligned} \int \frac{1}{x^2+1} \, dx=arctan(x)+c \newline \int \frac{-1}{x^2+1} \, dx=arccot(x)+c \end{aligned} \]

\[ \begin{aligned} \int \frac{1}{\sqrt{1-x^2}} \, dx=\arcsin (x)+c \newline = -\arccos(x)+c \newline \newline \int -\frac{1}{\sqrt{1-x^2}} \, dx=\arccos (x)+c \end{aligned} \]

三角函数：

\[ \int \cos (x) \, dx=\sin(x)+c \]

\[ \int \sin (x) \, dx=-\cos (x)+c \]

\[ \int \frac{1}{\cos ^2(x)} \, dx=\int \sec ^2(x) \, dx=\tan (x)+c \]

\[ \int \frac{1}{\sin ^2(x)} \, dx=\int \csc ^2(x) \, dx=-\cot (x)+c \]

\[ \int \tan (x) \sec (x) \, dx=\sec (x)+c \]

\[ \int \cot (x) \csc (x) \, dx=-\csc (x) \]

\[ \begin{aligned} \int \tan (x) \, dx=-\ln \vert \cos(x) \vert +c \end{aligned} \]

\[ \begin{aligned} \int \cot (x) \, dx=\ln \vert \sin(x) \vert +c \end{aligned} \]

\[ \begin{aligned} \int \csc (x) \, dx=\int \frac{1}{\sin(x)} \, dx=\ln \vert \csc(x) -\cot(x) \vert +c \end{aligned} \]

\[ \begin{aligned} \int \sec (x) \, dx=\int \frac{1}{\cos(x)} \, dx=\ln \vert \sec(x)+\tan(x)\vert +c \end{aligned} \]

幂函数

\[ \begin{aligned} \int a^x \, dx=\frac{a^x}{\ln (a)}+c \newline \int e^x \, dx=e^x+c \end{aligned} \]

二次项 或 根号

\[ \begin{aligned} \int \frac{1}{a^2+x^2} \, dx=\frac{\arctan \left(\frac{x}{a}\right)}{a}+c \end{aligned} \]

\[ \begin{aligned} \int \frac{1}{\sqrt{x^2+1}} \, dx=\ln \vert \sqrt{x^2+1}+x \vert +c \end{aligned} \]

\[ \begin{aligned} \int \frac{1}{\sqrt{a^2-x^2}} \, dx=\arcsin (\frac{x}{a})+c \end{aligned} \]

\[ \begin{aligned} \int \frac{1}{x^2-a^2} \, dx=-\frac{\tanh ^{-1}\left(\frac{x}{a}\right)}{a}=\frac{1}{2a} \ln \vert \frac{x-a}{x+a} \vert +c \end{aligned} \]

\[ \begin{aligned} \int \frac{1}{a^2-x^2} \, dx=\frac{\tanh ^{-1}\left(\frac{x}{a}\right)}{a}=\frac{1}{2a} \ln \vert \frac{x+a}{x-a} \vert +c \end{aligned} \]