Basic Integration Rules | Calculus Cheat Sheet

Power Rule for Integration

For $n \neq -1$ :

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

$\int x^{-1} \, dx = \int \frac{1}{x} \, dx = \ln|x| + C$

$\int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx$

$\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$

Integral	Result
$\int \frac{1}{\sqrt{1-x^2}} \, dx$	$\arcsin(x) + C$
$\int \frac{1}{1+x^2} \, dx$	$\arctan(x) + C$
$\int \frac{1}{\sqrt{x^2-1}} \, dx$	$\text{arcsec}(
$\int \frac{1}{\sqrt{a^2-x^2}} \, dx$	$\arcsin\left(\frac{x}{a}\right) + C$
$\int \frac{1}{a^2+x^2} \, dx$	$\frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$

$\int (4x^3 - 2x^2 + 5x - 3) \, dx = x^4 - \frac{2}{3}x^3 + \frac{5}{2}x^2 - 3x + C$

$\int (\sqrt{x} + \frac{1}{\sqrt{x}}) \, dx = \int (x^{1/2} + x^{-1/2}) \, dx$

$= \frac{x^{3/2}}{3/2} + \frac{x^{1/2}}{1/2} + C$

$= \frac{2}{3}x^{3/2} + 2x^{1/2} + C = \frac{2}{3}x\sqrt{x} + 2\sqrt{x} + C$

$\int \frac{x^2 + 1}{x^2} \, dx = \int \left(1 + \frac{1}{x^2}\right) dx = \int (1 + x^{-2}) \, dx$

$= x - \frac{1}{x} + C$

$\int (3\sin(x) - 2\cos(x)) \, dx = -3\cos(x) - 2\sin(x) + C$

$\int (e^{2x} - 3e^x) \, dx = \frac{1}{2}e^{2x} - 3e^x + C$

$\int \frac{1}{4 + x^2} \, dx = \int \frac{1}{2^2 + x^2} \, dx = \frac{1}{2}\arctan\left(\frac{x}{2}\right) + C$

$\int \frac{1}{\sqrt{3 - x^2}} \, dx = \int \frac{1}{\sqrt{(\sqrt{3})^2 - x^2}} \, dx = \arcsin\left(\frac{x}{\sqrt{3}}\right) + C$