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Mathematics Question on Definite Integral

Evaluate the limit : lim⁡x→π2(1(x−π2)3∫π2xcos⁡(1t3) dt)\lim_{x \to \frac{\pi}{2}} \left( \frac{1}{\left( x - \frac{\pi}{2} \right)^3} \int_{\frac{\pi}{2}}^x \cos \left( \frac{1}{t^3} \right) \, dt \right)

A

3π24\frac {3\pi^2}{4}

B

3π4\frac {3\pi}{4}

C

3π28\frac {3\pi^2}{8}

D

3π8\frac {3\pi}{8}

Answer

3π28\frac {3\pi^2}{8}

Explanation

Solution

Apply L’Hoˆpital’s Rule: The given expression is:lim⁡x→π2∫xπ2cos⁡(t2) dt(x−π2)3\text{Apply L'Hôpital's Rule: The given expression is:} \quad \lim_{x \to \frac{\pi}{2}} \frac{\int_{x}^{\frac{\pi}{2}} \cos \left( \frac{t}{2} \right) \, dt}{\left( x - \frac{\pi}{2} \right)^3}
Differentiate the Numerator and Denominator: Using the Fundamental Theorem of Calculus and L'Hôpital's Rule, we get:
=lim⁡x→π2x2cos⁡(x2)3(x−π2)2= \lim_{x \to \frac{\pi}{2}} \frac{x^2 \cos \left( \frac{x}{2} \right)}{3 \left( x - \frac{\pi}{2} \right)^2}
Evaluate the Expression as x→π2x \to \frac{\pi}{2}: As x→π2x \to \frac{\pi}{2}, substitute appropriate values and simplify the expression:
=3π28= \frac{3\pi^2}{8}
So, the correct option is: 3π28\frac{3\pi^2}{8}