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  
\[\int cos^3xdx=sinx - \frac{sin^3x}{3}+c\]
.

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