Table of elementary antiderivatives

Computing an integral almost always means recognising an antiderivative. It is therefore worth keeping in mind a small table of elementary antiderivatives, obtained simply by “reading the differentiation rules backwards”. The additive constant $C$ reminds us that the antiderivative is defined up to a constant (the indefinite integral).

In brief — Elementary antiderivatives

$f(x)$	$\mathcal{P}(f)$	$f(x)$	$\mathcal{P}(f)$
$x^n$ ($n\ne-1$)	$\dfrac{x^{n+1}}{n+1}+C$	$\cos x$	$\sin x + C$
$1$	$x+C$	$\sin x$	$-\cos x + C$
$\dfrac{1}{x}$	$\ln	x	+ C$
$\dfrac{1}{1+x^2}$	$\arctan x + C$	$\dfrac{1}{\cos^2 x}$	$\tan x + C$
$\dfrac{1}{\sqrt{1-x^2}}$	$\arcsin x + C$

The antiderivative is linear: $\mathcal{P}(\alpha f+\beta g) = \alpha\,\mathcal{P}(f)+\beta\,\mathcal{P}(g)$.

Thanks to linearity, an integral of a combination of functions splits into the sum of the individual integrals, each solvable with the table.

Links

Topics: Integral
Concepts: Indefinite integral · Linearity of the integral · Antiderivative
Skills: Integrating · Using formulae