∫ cosec^7x dx; x ∈ [0, 2π] =

Question

\(\int \limits_0^{2\pi} \mathrm{cosec}^7x \ dx =\)

(A) \(0\)

(B) \(1\)

(C) \(4\)

(D) \(2\pi\)

Answer 1

Correct option is (A) \(0\) 

We know, \(\int\limits_{0}^{2 a} {f}({x}) {d x}={0},\)  if \({f}({2 a}-{x})=-{f}({x})\)

Let \({f}({x})=\operatorname{cosec}^{7} {x}\)

Now, \({f}(2 \pi-{x})=\operatorname{cosec}^{7}(2 \pi-{x})=-\operatorname{cosec}^{7} {x}=-{f}({x})\) 

\(\therefore \int\limits_{0}^{2 \pi} \operatorname{cosec}^{7} {x} {d x}={0}\), 

Using the property \(\int\limits_{0}^{2 a} {f}({x}) {d x}={0}\), if \({f}(2 {a}-{x})=-{f}({x})\)