Integration Rules

Common Functions	Function	Integral
Constant	$\int adx$	$ax+C$
Variable	$\int xdx$	$\frac{x^2}{2}+C$
Square	$\int x^{2}dx$	$\frac{x^3}{3}+C$
Reciprocal	$\int \frac{1}{x}dx$	$\ln	x	+ C$
Exponential	$\int e^{x}dx$	$e^x+C$
	$\int a^{x}dx$	$\frac{a^x}{\ln (a)}+C$
	$\int \ln (x)dx$	$x\ln (x)-x+C$
Trigonometry (x in radians)	$\int \cos (x)dx$	$\sin (x)+C$
	$\int \sin (x)dx$	$-\cos (x)+C$
	$\int {\sec }^2(x)dx$	$\tan (x)+C$

Rules	Function	Integral
Multiplication by constant	$\int cf(x)dx$	$c\int f(x)dx$
Power Rule (n≠−1)	$\int x^{n}dx$	$\frac{x^{n+1}}{n+1}+C$
Sum Rule	$\int (f+g)dx$	$\int fdx+\int gdx$
Difference Rule	$\int (f-g)dx$	$\int fdx-\int gdx$

Examples

What is the integral of \sin{x}?

$\int \sin xdx=-\cos x+C$

What is the integral of 1/x?

$\int 1/xdx=ln|x|+C$

Power Rule

What is \int x^3 \,dx?

Use the Power Rule, where $n=3$: $\int x^{3}dx=\frac{x^{3+1}}{3+1}+C$ $\int x^{3}dx=\frac{x^4}{4}+C$

What is \int \sqrt{x} \,dx?

$\sqrt{x}$ is also $x^{0.5}$

Use the Power Rule, where $n=0.5$: $\int x^{3}dx=\frac{x^{0.5+1}}{0.5+1}+C$ $\int x^{3}dx=\frac{x^1.5}{1.5}+C$

Multiplication by constant

What is \int 6x^2 \,dx?

Move the 6 outside the integral: $\int 6x^{2}dx=6\int x^{2}dx$

Use the Power Rule on $x^2$: $\int 6x^{2}dx=6\frac{x^{2+1}}{2+1}+C$ $\int 6x^{2}dx=6\frac{x^3}{3}+C$

Simplify: $\int 6x^{2}dx=2x^3+C$

Sum Rule

What is \int (\cos{x} + x) \,dx?

Use the Sum Rule: $\int (\cos x+x)=\int cosxdx+\int xdx$

Work out the integral of each (using table above): $\int (\cos x+x)=\sin x+\frac{x^2}{2}+C$