Question

Evaluate the following definite integral :
${\int }_0^{\pi /2}$ $\frac{co{s}^{5}x}{si{n}^{5}x+co{s}^{5}x}dx$

Answer 1

Consider the given integral.

$I={\int }_0^{\pi /2}\frac{{\cos }^{5}x}{{\sin }^{5}x+{\cos }^{5}x}dx$ ……. (1)

We know that

${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$

Therefore,

$I={\int }_0^{\pi /2}\frac{{\sin }^{5}x}{{\cos }^{5}x+{\sin }^{5}x}dx$ ……. (2)

On adding equation (1) and (2), we get

$I={\int }_0^{\pi /2}\frac{{\sin }^{5}x+{\cos }^{5}x}{{\cos }^{5}x+{\sin }^{5}x}dx$

$I={\int }_0^{\pi /2}1dx$

$I={\left[x\right]}_0^{\pi /2}$

$I=\frac{\pi }{2}-0$

$I=\frac{\pi }{2}$

Hence, this is the answer.