Question

By using the properties of definite integrals,evaluate the integrals: $\int^\frac{\pi}{2}_0 \frac {{sin^{\frac32}\ x\ dx}}{{sin^{\frac32}\ x+cos^{\frac32}\ x}} \, dx$

Answer 1

Answer:

Step-by-step explanation:

concept:

$\int\limits^{\frac{\pi}{2}}_0{f(x)\:dx=\int\limits^{\frac{\pi}{2}}_0{f(\frac{\pi}{2}-x)\:dx$

Let,

$I=\int\limits^{\frac{\pi}{2}}_0{\frac{sin^{\frac{3}{2}}x}{sin^{\frac{3}{2}}x+cos^{\frac{3}{2}}x}\:dx.............(1)$

By properties of definite integral,

$I=\int\limits^{\frac{\pi}{2}}_0{\frac{sin^{\frac{3}{2}}(\frac{\pi}{2}-x)}{sin^{\frac{3}{2}}(\frac{\pi}{2}-x)+cos^{\frac{3}{2}}(\frac{\pi}{2}-x)}\:dx$

$I=\int\limits^{\frac{\pi}{2}}_0{\frac{cos^{\frac{3}{2}}x}{cos^{\frac{3}{2}}x+sin^{\frac{3}{2}}x}\:dx............(2)$

Adding (1) and (2) we get

$2I=\int\limits^{\frac{\pi}{2}}_0[\frac{sin^{\frac{3}{2}}x}{sin^{\frac{3}{2}}x+cos^{\frac{3}{2}}x}+{\frac{cos^{\frac{3}{2}}x}{cos^{\frac{3}{2}}x+sin^{\frac{3}{2}}x}]\:dx$

$2I=\int\limits^{\frac{\pi}{2}}_0[\frac{sin^{\frac{3}{2}}x+cos^{\frac{3}{2}}x}{sin^{\frac{3}{2}}x+cos^{\frac{3}{2}}x}]\:dx$

$2I=\int\limits^{\frac{\pi}{2}}_0{1}\:dx\\\\2I=[x]^{\frac{\pi}{2}}_0\\\\2I=\frac{\pi}{2}\\\\I=\frac{\pi}{4}$