Question

Evaluate $\stackrel{\pi /2}{\underset{0}{\int }}\frac{{\sin }^{5}x}{{\sin }^{5}x+{\cos }^{5}x}dx$

Answer 1

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

As we know that,

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

Therefore,

$I={\int }_0^{\frac{\pi }{2}}\frac{{\sin }^5(\frac{\pi }{2}+0-x)}{{\sin }^5(\frac{\pi }{2}+0-x)+{\cos }^5(\frac{\pi }{2}+0-x)}dx$

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

Adding equation $\left(1\right)\&\left(2\right)$, we have

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

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

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

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

$\Rightarrow I=\frac{1}{2}{\left[x\right]}_0^{\frac{\pi }{2}}$

$\Rightarrow I=\frac{\pi }{4}$

Hence ${\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{5}x}{{\sin }^{5}x+{\cos }^{5}x}dx=\frac{\pi }{4}$