TITLE: Evaluating $\int \cos^4(x)\operatorname d\!x$
QUESTION [6 upvotes]: I want to Evaluate integral :$$\int \cos^4(x)\operatorname d\!x$$ And think about doing the following thing: 
$$
\int \left(1-\sin^2(x)\right)^2\operatorname d\!x
 \to \int \left(1-2\sin^2(x)+\sin^4(x)\right)\operatorname d\!x
$$ but I think I just complicated it.
Any suggestions? Thanks!

REPLY [3 votes]: Using  the reduction formulae,
$$\int\cos^nxdx=\frac{\cos^{n-1}x\sin x}n+\frac{n-1}n \int\cos^{n-2}xdx$$
Putting $n=2,$ $$\int\cos^2xdx=\frac{\cos x\sin x}2+\frac12 \int dx=\frac{\cos x\sin x}2+\frac12 x+C$$
Putting $n=4,$ $$\int\cos^4xdx=\frac{\cos^3x\sin x}4+\frac34 \int\cos^2xdx$$