lim x 2 ( x 3 ( / 2 ) 3 ( sin ( 2 t 1 / 3 ) + cos ( t 1 / 3 ) ) ?d t ( x - 2 ) 2 ) is equal to [2024]

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Solution

Q.
$\lim_{x \to \frac{π}{2}} (\frac{\int_{x^{3}}^{{(π / 2)}^{3}} (\sin (2 t^{1 / 3}) + \cos (t^{1 / 3})) d t}{{(x - \frac{π}{2})}^{2}})$ is equal to [2024]

1 $\frac{11 π^{2}}{10}$

2 $\frac{3 π^{2}}{2}$

3 $\frac{5 π^{2}}{9}$

4 $\frac{9 π^{2}}{8}$

Ans.
(4)

Let $L = \lim_{x \to π / 2} [\frac{\int_{x^{3}}^{{(π / 2)}^{3}} (\sin (2 t^{1 / 3}) + \cos (t^{1 / 3})) d t}{{(x - \frac{π}{2})}^{2}}]$ $\frac{0}{0} form, so by L'Hospital rule$

$= \lim_{x \to \frac{π}{2}} \frac{\frac{d}{d x} \int_{x^{3}}^{{(π / 2)}^{3}} (\sin (2 t^{1 / 3}) + \cos (t^{1 / 3})) d t}{2 (x - \frac{π}{2})}$ $\sin (2 \times \frac{π}{2}) \cdot \frac{d}{d x} {(\frac{π}{2})}^{3} - \sin (2 x) \cdot \frac{d}{d x} x^{3}$

$= \lim_{x \to π / 2} \frac{+ (\cos \frac{π}{2} \frac{d}{d x} {(\frac{π}{2})}^{3} - \cos x \cdot \frac{d}{d x} x^{3})}{2 (x - \frac{π}{2})}$ (By Leibnitz rule of integration)

$= \lim_{x \to π / 2} \frac{- 3 x^{2} \sin 2 x - 3 x^{2} \cos x}{2 (x - \frac{π}{2})} (\frac{0}{0} form)$

$= \lim_{x \to π / 2} \frac{- 6 x \sin 2 x - 6 x^{2} \cos 2 x - 6 x \cos x + 3 x^{2} \sin x}{2}$

$= \frac{6 \frac{π^{2}}{4} + \frac{3 π^{2}}{4}}{2} = \frac{9 π^{2}}{8}$

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