_0^ /2 ^3 ,d = - Vidyadip

MCQ

$\int_0^{\pi /2} {\sqrt {\cos \theta } {{\sin }^3}\theta } \,d\theta = $

A
$\frac{{20}}{{21}}$
✓
$\frac{8}{{21}}$
C
$\frac{{ - 20}}{{21}}$
D
$\frac{{ - 8}}{{21}}$

✓

Answer

Correct option: B.

$\frac{8}{{21}}$

b
(b) Let $I = \int_0^{\pi /2} {\sqrt {\cos \theta } } {\sin ^3}\theta \,\,d\theta $

Put $t = \cos \theta \Rightarrow dt = - \sin \theta \,\,d\theta ,$ then

$I =$$ - \int_1^0 {{t^{1/2}}(1 - {t^2})dt = \int_0^1 {({t^{1/2}} - {t^{5/2}})} } $$dt$

$I = $$\left[ {\frac{2}{3}{t^{3/2}} - \frac{2}{7}{t^{7/2}}} \right]_0^1 = \frac{8}{{21}}$.

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