The value of _0^ /2 ^3 ,x ,x , + , ,x ,dx is - Vidyadip

MCQ

The value of $\int\limits_0^{\pi /2} {\frac{{{{\sin }^3}\,x}}{{\sin \,x\, + \,\cos \,x}}} \,dx$ is

A
$\frac{{\pi \, - \,2}}{4}$
B
$\frac{{\pi \, - \,1}}{2}$
✓
$\frac{{\pi \, - \,1}}{4}$
D
$\frac{{\pi \, - \,2}}{8}$

✓

Answer

Correct option: C.

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

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

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

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

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

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

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

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