_ x 2 (1 (x- 2 )^2 _ x^2 ^ ( 2 )^3 (1 t^3 ) dt ) is equal to - Vidyadip

MCQ

$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2} \int_{x^2}^{\left(\frac{\pi}{2}\right)^3} \cos \left(\frac{1}{t^3}\right) dt \right)$ is equal to

A
$\frac{3 \pi}{8}$
B
$\frac{3 \pi^2}{4}$
C
$\frac{3 \pi^2}{8}$
D
$\frac{3 \pi}{4}$

✓

Answer

Using L'hopital rule
$ =\lim _{x \rightarrow \frac{\pi^{-}}{2}} \frac{0-\cos x \times 3 x^2}{2\left(x-\frac{\pi}{2}\right)}$
$=\lim _{x \rightarrow \frac{\pi^{-}}{2}} \frac{\sin \left(x-\frac{\pi}{2}\right)}{2\left(x-\frac{\pi}{2}\right)} \times \frac{3 \pi^2}{4}$
$=\frac{3 \pi^2}{8} $

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