Solve this following

Question:

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

 

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

  2. $\frac{\pi-2}{8}$

  3. $\frac{\pi-1}{4}$

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


Correct Option: , 3

Solution:

$I=\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin x+\cos x} d x$

$\Rightarrow I=\int_{0}^{\pi / 4} \frac{\sin ^{3} x+\cos ^{3} x}{\sin x+\cos x} d x$

$=\int_{0}^{\pi / 4}(1-\sin x \cos x) d x$

$=\left(x-\frac{\sin ^{2} x}{2}\right)_{0}^{\pi / 4}$

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

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

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