_0^ / 4 ^2 x ^2 x ( ^3 x+ ^3 x )^2 d x is equal to - Vidyadip

MCQ

$\int_0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x$ is equal to

A
$1 / 12$
B
$1 / 9$
✓
$1 / 6$
D
$1 / 3$

✓

Answer

Correct option: C.

$1 / 6$

c
Divide $\mathrm{Nr} \& \mathrm{Dr}$ by $\cos \mathrm{x}$

$\int_0^{\pi / 4} \frac{\tan ^2 x \sec ^2 x d x}{\left(1+\tan ^3 x\right)^2} d x$

Let $1+\tan ^3 \mathrm{x}=\mathrm{t}$

$\tan ^2 \mathrm{x} \sec ^2 \mathrm{x} d \mathrm{x}=\frac{\mathrm{dt}}{3}$

$\frac{1}{3} \int_1^2 \frac{\mathrm{dt}}{\mathrm{t}^2}=\frac{1}{6}$

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