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Table of integrals: standard integrals and integration rules

The standard integrals and basic integration methods, on one page. The integral is on the left, the antiderivative on the right.

Methods: by parts, substitution, linear argument

Integral	Result
$\int u d v$	$u v - \int v d u$
$\int f (g (x)) g^{'} (x) d x$	$\int f (u) d u, u = g (x)$
$\int f (a x + b) d x$	$\frac{1}{a} F (a x + b) + C$

Powers, exponentials and logarithms

Integral	Result
$\int x^{n} d x$	$\frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1$
$\int \frac{1}{x} d x$	$ln ∣ x ∣ + C$
$\int e^{x} d x$	$e^{x} + C$
$\int a^{x} d x$	$\frac{a ^{x}}{ln a} + C$
$\int ln x d x$	$x ln x - x + C$

Trigonometric

Integral	Result
$\int sin x d x$	$- cos x + C$
$\int cos x d x$	$sin x + C$
$\int tan x d x$	$- ln ∣ cos x ∣ + C$
$\int \frac{1}{cos ^{2} x} d x$	$tan x + C$
$\int \frac{1}{sin ^{2} x} d x$	$- cot x + C$

Integrals giving arctan, arcsin and logarithms

Integral	Result
$\int \frac{1}{1 + x ^{2}} d x$	$arctan x + C$
$\int \frac{1}{a ^{2} + x ^{2}} d x$	$\frac{1}{a} arctan \frac{x}{a} + C$
$\int \frac{1}{1 - x ^{2}} d x$	$arcsin x + C$
$\int \frac{1}{a ^{2} - x ^{2}} d x$	$arcsin \frac{x}{a} + C$
$\int \frac{1}{x ^{2} - a ^{2}} d x$	$\frac{1}{2 a} ln \frac{x - a}{x + a} + C$
$\int \frac{1}{x ^{2} + a} d x$	$ln x + x^{2} + a + C$

Notes

A quick check for any result: differentiate it, and you should get back the function inside the integral.

For integration by parts, pick a u whose derivative is simpler and a dv that is easy to integrate. In ∫x·e^x dx, choose u = x.