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