Question:
The value of $\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin x+\cos x} d x$ is
Correct Option: , 3
Solution:
$I=\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin x+\cos x} d x$
$\Rightarrow I=\int_{0}^{\pi / 4} \frac{\sin ^{3} x+\cos ^{3} x}{\sin x+\cos x} d x$
$=\int_{0}^{\pi / 4}(1-\sin x \cos x) d x$
$=\left(x-\frac{\sin ^{2} x}{2}\right)_{0}^{\pi / 4}$
$=\frac{\pi}{4}-\frac{1}{4}$
$=\frac{\pi-1}{4}$