Question:
$\int_{0}^{2 \pi} \cos ^{5} x d x$
Solution:
Let $I=\int_{0}^{2 \pi} \cos ^{5} x d x$ ...(1)
$\cos ^{5}(2 \pi-x)=\cos ^{5} x$
It is known that,
$\int_{0}^{2 a} f(x) d x=2 \int_{0}^{a} f(x) d x$, if $f(2 a-x)=f(x)$
$=0$ if $f(2 a-x)=-f(x)$
$\therefore I=2 \int_{0}^{\pi} \cos ^{5} x d x$
$\Rightarrow I=2(0)=0$ $\left[\cos ^{5}(\pi-x)=-\cos ^{5} x\right]$