Definite Integration
Properties of Definite Integrals
IIT-JEE 2006
Grade 12
Question:
$\int_{-\pi/2}^{\pi/2} \left[ x^2 + \ln\left(\frac{\pi + x}{\pi - x}\right) \right] \cos x \, dx$ equals:
(1) $\frac{\pi^2}{2} - 4$
(2) $\frac{\pi^2}{2} + 4$
(3) $\pi^2 - 4$
(4) $\frac{\pi^2}{4} - 2$
Step-by-Step Solution
Key Concept: Check whether each part of the integrand is an even or an odd function on the symmetric interval $[-\pi/2, \pi/2]$ — the odd part contributes nothing to the integral.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)