## Table of Integrals

## Handy Table of Integrals

### Power of *x*

\begin{align*}

\int a \, dx &= ax + C \\ \\

\int x^n \, dx &= \frac{x^{n+1}}{n+1} + C \qquad n \ne -1 \\ \\

\int \frac{1}{x} \, dx &= \ln |x| + C \\

\end{align*}

### Exponential and Logarithmic

\begin{align*}

\int e^x \, dx &= e^x + C \\ \\

\int a^x \, dx &= \frac{a^x}{\ln a} + C \\ \\

\int \ln x \, dx &= x \ln x – x + C

\end{align*}

### Trigonometric

\begin{align*}

\int \sin x \, dx &= -\cos x + C &&& \int \csc x \, dx &= \ln |\csc x – \cot x| + C \\ \\

\int \cos x \, dx &= \sin x + C &&& \int \sec x \, dx &= \ln |\sec x + \tan x| + C \\ \\

\int \tan x \, dx &= \ln |\sec x| + C &&& \int \cot x \, dx &= \ln |\sin x| + C \\

&= -\ln |\cos x| + C \\ \\

\int \sec^2 x \, dx &= \tan x + C &&& \int \csc^2 x \, dx &= -\cot x + C \\ \\

\int \sec x \tan x \, dx &= \sec x + C &&& \int \csc x \cot x \, dx &= – \csc x + C \\ \\

\end{align*}

\begin{align*}

\int \sin^2 x \, dx &= \int \left(\frac{1}{2} – \frac{1}{2}\cos 2x \right) \, dx = \frac{1}{2}x – \frac{1}{4}\sin 2x + C \\ \\

\int \cos^2 x \, dx &= \int \left( \frac{1}{2} + \frac{1}{2}\cos 2x \right) \, dx = \frac{1}{2}x + \frac{1}{4}\sin 2x + C \\ \\

\int \tan^2 x \, dx &= \int \left(\sec^2x – 1\right)\, dx = \tan x – x + C

\end{align*}

### Inverse Trigonometric

\begin{align*}

\int \frac{dx}{\sqrt{a^2 – x^2}} &= \sin^{-1}\frac{x}{a} + C \\ \\

\int \frac{dx}{a^2 + x^2} &= \frac{1}{a} \tan^{-1}\frac{x}{a} + C \\ \\

\int \frac{dx}{x\sqrt{x^2 – a^2}} &= \frac{1}{a} \sec^{-1} \left|\frac{x}{a}\right| + C

\end{align*}

### Hyperbolic

\begin{align*}

\text{Reminder:} \\

\sinh x &= \frac{e^x – e^{-x}}{2} & \cosh x &= \frac{e^x + e^{-x}}{2} & \tanh x &= \frac{\sinh x}{\cosh x} \\ \\

\text{csch }x &= \frac{1}{\sinh x} & \text{sech }x &= \frac{1}{\cosh x} & \coth x &= \frac{\cosh x}{\sinh x} \\

\end{align*}

\begin{align*}

\int \sinh x \, dx &= \cosh x + C &&& \int \text{csch }x \, dx &= \ln \left|\tanh \frac{1}{2}x\right| + C \\ \\

\int \cosh x \, dx &= \sinh x + C &&& \int \text{sech }x \, dx &= \tan^{-1}|\sinh x| + C \\ \\

\int \tanh x \, dx &= \ln \cosh x + C &&& \int \coth x \, dx &= \ln |\sinh x| + C

\end{align*}

*Tip:* You can integrate any function, for free,

using *WolframAlpha’s* Integral Calculator.