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STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2} \int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \cos \left(\frac{1}{t^3}\right) d t\right)$ is equal to
  • A
     $\frac{3 \pi}{8}$
  • B
    $\frac{3 \pi^2}{4}$
  • ✓
    $\frac{3 \pi^2}{8}$
  • D
     $\frac{3 \pi}{4}$
✓

Answer

Correct option: C.
$\frac{3 \pi^2}{8}$
c
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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