Question

By using the properties of definite integrals, evaluate the integral: $\int^{\frac{π}{2}}_0 \frac{sin^{\frac{3}{2}}xdx}{sin^{\frac{3}{2}}x+cos^{\frac{3}{2}}x}$

Answer 1

The Correct Answer is:$I = \frac{\pi}{4}$
Let $I = \int^{\frac{π}{2}}_0 \frac{sin^{\frac{3}{2}}xdx}{sin^{\frac{3}{2}}x+cos^{\frac{3}{2}}x} ...(1)$
⇒$I = \int^{\frac{π}{2}}_0 \frac{sin^{\frac{3}{2}}(\frac{\pi}{2} - x)}{sin^{\frac{3}{2}}(\frac{\pi}{2} - x)+cos^{\frac{3}{2}}(\frac{\pi}{2} - x)} dx \,\,\,\,\,\,\, (\int^a_0 f(x)dx= \int^a_0 f(a-x)dx)$
⇒$I = \int^{\frac{π}{2}}_0 \frac{cos^{\frac{3}{2}}x}{sin^{\frac{3}{2}}x+cos^{\frac{3}{2}}x}dx...(2)$
Adding(1)and(2),we obtain
$2I = \int^{\frac{π}{2}}_0 \frac{sin^{\frac{3}{2}}x+cos^{\frac{3}{2}}x}{sin^{\frac{3}{2}}x+cos^{\frac{3}{2}}x}dx$
⇒$2I = \int^{\frac{π}{2}}_0 1.dx$
⇒$2I = [x]^\frac{\pi}{2}_0$
⇒$2I = \frac{\pi}{2}$
⇒$I = \frac{\pi}{4}$