_0^ 2 cosec^7 x d x= - Vidyadip

Question

$\int_0^{2 \pi} \operatorname{cosec}^7 x d x=$

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Answer

We know, $\int_0^{2 a} f(x) d x=0$, if $f(2 a-x)=-f(x)$
Let $f(x)=\operatorname{cosec}^7 x$
Now, $f(2 \pi-x)=\operatorname{cosec}^7(2 \pi-x)=-\operatorname{cosec}^7 x=-f(x)$
$\therefore \int_0^{2 \pi} \operatorname{cosec}^7 x d x=0 ;$ Using the property $\int_0^{2 a} f(x) d x=0$, if $f(2 a-x)=-f(x)$

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