# Standard Integrals

Where appropriate $a$, $b$, $C$ and $n$ are constants. Note that $a\ne 0$. Also $u$ and $v$ are integrable functions of $x$.

### Basic Rules

$$\newcommand{\sech}{\operatorname{sech}} \newcommand{\cosech}{\operatorname{cosech}} \newcommand{\cosec}{\operatorname{cosec}} \newcommand{\ctext}[1]{\style{font-family:Arial}{\text{#1}}} \int{x^n\:dx=\frac{x^{n+1}}{n+1}}+C,\qquad n\ne -1$$

$$\int{dx}=x+C$$

$$\int{\left(ax+b\right)^n\:dx}=\frac{\left(ax+b\right)^{n+1}}{a(n+1)}+C\quad n\ne -1$$

$$\int{au+bv\:dx}=a\int{u\:dx}+b\int{v\:dx}$$

### Exponential and Logarithmic Functions

$$\int{\frac{1}{x}\:dx}=\ln |x|+C$$

$$\int{\frac{dx}{ax+b}}=\frac{1}{a}\ln |ax+b|+C$$

$$\int{e^{ax+b}\:dx}=\frac{1}{a}e^{ax+b}+C$$

$$\int{a^x\:dx}=\frac{1}{\ln a}a^x+C,\qquad a>0, a\ne 1$$

### Trigonometric Functions

$$\int{\cos\left( ax+b\right)\:dx}=\frac{1}{a}\sin\left(ax+b\right) +C$$

$$\int{\sin\left( ax+b\right)\:dx}=-\frac{1}{a}\cos\left(ax+b\right) +C$$

$$\int{\tan\left( ax+b\right)\:dx}=\frac{1}{a}\ln |\sec\left(ax+b\right)|+C$$

$$\int{\sec\left( ax+b\right)\:dx}=\frac{1}{a}\ln |\sec\left(ax+b\right)+\tan\left( ax+b\right)|+C$$

$$\int{\cosec\left( ax+b\right)\:dx}=\frac{1}{a}\ln |\cosec\left(ax+b\right)-\cot\left( ax+b\right)|+C$$

$$\int{\cot\left( ax+b\right)\:dx}=\frac{1}{a}\ln |\sin\left(ax+b\right)|+C$$

$$\int{\sec^2\left( ax+b\right)\:dx}=\frac{1}{a}\tan\left(ax+b\right)+C$$

$$\int{\cosec^2\left( ax+b\right)\:dx}=-\frac{1}{a}\cot\left(ax+b\right)+C$$

$$\int{\sec\left( ax+b\right)\tan\left( ax+b\right)\:dx}=\frac{1}{a}\sec\left(ax+b\right)+C$$

$$\int{\cosec\left( ax+b\right)\cot\left( ax+b\right)\:dx}=-\frac{1}{a}\cosec\left(ax+b\right)+C$$

### Inverse Trigonometric Functions

$$\int{\frac{dx}{\sqrt{a^2-x^2}}}=\sin^{-1}\left(\frac{x}{a}\right)+C,\quad x^2<a^2$$

$$\int{\frac{dx}{a^2+x^2}}=\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C,\quad a\ne 0$$

$$\int{\frac{dx}{x\sqrt{x^2-a^2}}}=\frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right)+C,\quad x^2>a^2$$

### Hyperbolic Functions

$$\int{\cosh\left( ax+b\right)\:dx}=\frac{1}{a}\sinh\left(ax+b\right) +C$$

$$\int{\sinh\left( ax+b\right)\:dx}=\frac{1}{a}\cosh\left(ax+b\right) +C$$

$$\int{\tanh\left( ax+b\right)\:dx}=\frac{1}{a}\ln |\cosh\left(ax+b\right)|+C$$

$$\int{\coth\left( ax+b\right)\:dx}=\frac{1}{a}\ln |\sinh\left(ax+b\right)|+C$$

$$\int{\sech^2\left( ax+b\right)\:dx}=\frac{1}{a}\tanh\left(ax+b\right)+C$$

$$\int{\cosech^2\left( ax+b\right)\:dx}=-\frac{1}{a}\coth\left(ax+b\right)+C$$

$$\int{\sech\left( ax+b\right)\tanh\left( ax+b\right)\:dx}=\frac{1}{a}\sech\left(ax+b\right)+C$$

$$\int{\cosech\left( ax+b\right)\coth\left( ax+b\right)\:dx}=-\frac{1}{a}\cosech\left(ax+b\right)+C$$

### Inverse Hyperbolic Functions

$$\int{\frac{dx}{\sqrt{a^2+x^2}}}=\sinh^{-1}\left(\frac{x}{a}\right)+C,\quad a>0$$

$$\int{\frac{dx}{\sqrt{x^2-a^2}}}=\cosh^{-1}\left(\frac{x}{a}\right)+C,\quad0<a<x$$

\begin{align*}\int{\frac{dx}{a^2-x^2}}&=\frac{1}{a}\tanh^{-1}\left(\frac{x}{a}\right)+C,\quad x^2<a^2\cr &=\frac{1}{a}\coth^{-1}\left(\frac{x}{a}\right)+C,\quad x^2>a^2\end{align*}

$$\int{\frac{dx}{x\sqrt{a^2-x^2}}}=\frac{1}{a}\sech^{-1}\left(\frac{x}{a}\right)+C,\quad 0<x<a$$

$$\int{\frac{dx}{x\sqrt{a^2+x^2}}}=-\frac{1}{a}\cosech^{-1}\left|\frac{x}{a}\right|+C,\quad x\ne 0$$