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

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Solution

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

1 $\frac{3 π^{2}}{4}$

2 $\frac{3 π^{2}}{8}$

3 $\frac{3 π}{8}$

4 $\frac{3 π}{4}$

Ans.
(2)

Let $I = \lim_{x \to \frac{π}{2}} (\frac{1}{{(x - \frac{π}{2})}^{2}} \int_{x^{3}}^{{(\frac{π}{2})}^{3}} \cos (t^{1 / 3}) d t)$ $(\frac{0}{0} form)$

By Leibnitz theorem,

$\frac{d}{d x} \int_{h (x)}^{g (x)} f (t) d t = f (g (x)) \times g' (x) - f (h (x)) \times h' (x)$

$∴$ By applying L'Hospital rule, we get

$I = \lim_{x \to \frac{π}{2}} (\frac{- \cos x (3 x^{2})}{2 (x - \frac{π}{2})})$

$= \lim_{x \to \frac{π}{2}} (\frac{3 x^{2} \sin x - 6 x \cos x}{2}) (Applying L'Hospital rule)$

$= \frac{3 π^{2}}{8}$

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