Question

By using properties of definite integrals, evaluate the integrals
${\int }_0^{\frac{\pi }{2}}\frac{co{s}^{5}x}{si{n}^{5}x+co{s}^{5}x}dx.$

Answer 1

I $={\int }_0^{\frac{\pi }{2}}\frac{co{s}^{5}x}{si{n}^{5}x+co{s}^{5}x}dx..........\left(i\right)\Rightarrow I={\int }_0^{\frac{\pi }{2}}\frac{co{s}^5(\frac{\pi }{2}-x)}{si{n}^5(\frac{\pi }{2}-x)+co{s}^5(\frac{\pi }{2}-x)}[\because {\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x\left)dx\right]={\int }_0^{\frac{\pi }{2}}\frac{si{n}^{5}x}{co{s}^{5}x+si{n}^{5}x}dx.........\left(ii\right)$
On adding Eqs. (i) and (ii) we get
$\text{2I =}$ ${\int }_0^{\frac{\pi }{2}}\frac{co{s}^{5}x+si{n}^{5}x}{si{n}^{5}x+co{s}^{5}x}dx={\int }_0^{\frac{\pi }{2}}1dx=\frac{\pi }{2}-0\to I=\frac{\pi }{4}$