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$\int_{0}^{2 \pi} \cos ^{5} x d x$


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 $f(2 a-x)=-f(x)$

$\therefore I=2 \int_{0}^{\pi} \cos ^{5} x d x$

$\Rightarrow I=2(0)=0$                         $\left[\cos ^{5}(\pi-x)=-\cos ^{5} x\right]$


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