Q. The value of $\int_{0}^{\pi/2}\frac{\sin^{2}{x - \cos^{2}x}}{\sin^{3}{x + \cos^{3}x}}dx$ is
A) 0
B) 1
C) 2
D) 3
Explanation
(a)
Let $I = \int_{0}^{\pi/2}\frac{(\sin^{2}{x - \cos^{2}{x)dx}}}{(\sin{x + \cos{x)(\sin^{2}{+ {cos^{2}}{x - \sin{x\cos{x)}}}}}}}$
$I = \int_{0}^{\pi/2}\frac{(\sin{x - \cos{x)dx}}}{(1 - \sin{x\cos\ x)}}$
$I = \int_{0}^{\pi/2}\frac{\cos{x - \sin x}}{1 - \sin{x\cos\ x}}dx\ \ \ \ \left( \because\int_{0}^{a}{f(x) = \int_{0}^{a}{f(a - x)}} \right)$
$\Rightarrow \ I = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.