# By using the properties of definite integrals,evaluate the integral$\int\limits_0^\frac{\pi}{2}\frac{\sin^\frac{3}{2}\;x\;dx}{\sin^\frac{3}{2}\;x+\cos^\frac{3}{2}\;x}$

Toolbox:
• (i)$\int\limits_0^af(x)dx=\int\limits_0^af(a-x)dx$
• (ii)$\sin(\frac{\pi}{2}-x)=\cos x$
• (iii)$\cos (\frac{\pi}{2}-x)=\sin x$
Given $\int\limits_0^\frac{\pi}{2}\large\frac{\sin^\frac{3}{2}\;x\;}{\sin^\frac{3}{2}\;x+\cos^\frac{3}{2}\;x}dx-----(1)$

By applying property $\int\limits_0^af(x)dx=\int\limits_0^af(a-x)dx$

$I=\int\limits_0^\frac{\pi}{2}\large\frac{\sin^\frac{3}{2}\;(\frac{\pi}{2}-x)\;}{\sin^\frac{3}{2}\;(\frac{\pi}{2}-x)+\cos^\frac{3}{2}\;(\frac{\pi}{2}-x)}$

But $\sin(\frac{\pi}{2}-x)=\cos x$ and

$\cos(\frac{\pi}{2}-x)=\sin x$

Therefore $I=\int\limits_0^\frac{\pi}{2}\large\frac{\cos^\frac{3}{2}\;x\;}{\cos^\frac{3}{2}\;x+\sin^\frac{3}{2}\;x}dx-----(2)$

$2I=\int\limits_0^\frac{\pi}{2}\large\frac{\sin^{\frac{3}{2}}\;x+\cos ^{\frac{3}{2}}x\;}{\sin^\frac{3}{2}\;x+\cos^\frac{3}{2}\;x}dx$

$2I=\int \limits_0^{\frac{\pi}{2}}dx$

On integrating,

$2I=[x]_0^{\frac{\pi}{2}}$

On applying limits

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

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

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