The value of
Question:

The value of $\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin x+\cos x} d x$ is:

1. (1) $\frac{\pi-2}{8}$

2. (2) $\frac{\pi-1}{4}$

3. (3) $\frac{\pi-2}{4}$

4. (4) $\frac{\pi-1}{2}$

Correct Option: , 2

Solution:

Let $I=\int_{0}^{\pi / 2} \frac{\sin ^{3} x d x}{\sin x+\cos x}$…..(1)

Use the property $\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x$

$\Rightarrow I=\int_{0}^{\pi / 2} \frac{\cos ^{3} x d x}{\sin x+\cos x}$ …..(2)

Adding equation (1) and (2), we get

$2 I=\int_{0}^{\pi / 2}\left(1-\frac{1}{2} \sin (2 x)\right) d x$

$\Rightarrow I=\frac{1}{2}\left[x+\frac{1}{4} \cos 2 x\right]_{0}^{\pi / 2}$

$\Rightarrow I=\frac{\pi-1}{4}$