_ x x 2 ( _ x^3 ^ ( / 2)^3 ( (2 t^ 1 / 3 )+ (t^ 1 / 3 ) ) d t (x-… - Vidyadip

MCQ

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

A
$\frac{9 \pi^2}{8}$
B
$\frac{11 \pi^2}{10}$
C
$\frac{3 \pi^2}{2}$
D
$\frac{5 \pi^2}{9}$

✓

Answer

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{0-\{\sin (2 x)+\cos (x)\} \cdot 3 x^2}{2\left(x-\frac{\pi}{2}\right)}$
$=\lim _{x \rightarrow \frac{\pi}{2}} \frac{-\{2 \sin x \cos x+\cos x\} 3 x^2}{2\left(x-\frac{\pi}{2}\right)}$
$=\lim _{x \rightarrow \frac{\pi}{2}}\left\{\frac{2 \sin x \sin \left(\frac{\pi}{2}-x\right)}{2\left(x-\frac{\pi}{2}\right)}+\frac{\sin \left(\frac{\pi}{2}-x\right)}{2\left(\frac{\pi}{2}-x\right)}\right\} 3 x^2$
$=\left(1(1)+\frac{1}{2}\right) 3\left(\frac{\pi}{2}\right)^2$
$=\frac{9 \pi^2}{8}$

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