## Integration By Substitution

Integration by substitution is a commonly used technique of integration.
Example:
$\in cos^3 x dx$
.
First write as
$\int cos x cos^2 x dx= \int cos x(1-sin^2x)dx = \int cos x - cos x sin^2xdx$

The first term in the integrand -
$cosx$
- integrates to give
$sinx$
. To find
$\int cosxsin^2x dx$
substitute
$u=sinx$
then
$\frac{du}{dx}=cosx \rightarrow du=cosxdx \rightarrow dx=\frac{du}{cosx}$
, The integral becomes
$\int cosxu^2 \frac{du}{cosx} du = \int u^2du = \frac{u^3}{3}+c$
. Replace
$u$
by
$sinx$
to get
$\frac{sin^3x}{3} +c$
.
Hence
$\int cos^3xdx=sinx - \frac{sin^3x}{3}+c$
.