JEE Mains · Maths · STD 12 - 7.2 definite integral
\(\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2} \int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \cos \left(\frac{1}{t^3}\right) d t\right)\) is equal to
- A \(\frac{3 \pi}{8}\)
- B \(\frac{3 \pi^2}{4}\)
- C \(\frac{3 \pi^2}{8}\)
- D \(\frac{3 \pi}{4}\)
Answer & Solution
Correct Answer
(C) \(\frac{3 \pi^2}{8}\)
Step-by-step Solution
Detailed explanation
Using L'hopital rule \( =\lim _{x \rightarrow \frac{\pi^{-}}{2}} \frac{0-\cos x \times 3 x^2}{2\left(x-\frac{\pi}{2}\right)} \) \( =\lim _{x \rightarrow \frac{\pi^{-}}{2}} \frac{\sin \left(x-\frac{\pi}{2}\right)}{2\left(x-\frac{\pi}{2}\right)} \times \frac{3 \pi^2}{4}\)…
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