Definite Integration
Integration by Substitution
IIT-JEE 1997
Grade 12
Question:
$\int_0^1 \frac{\tan^{-1} x}{1 + x^2} \, dx$ equals:
(1) $\frac{\pi^2}{32}$
(2) $\frac{\pi^2}{16}$
(3) $\frac{\pi^2}{8}$
(4) $\frac{\pi}{4}$
Step-by-Step Solution
Key Concept: Substitute $t = \tan^{-1} x$ so that $dt = \frac{dx}{1 + x^2}$; this converts the integral into a simple integral of $t$ with new limits $0$ and $\pi/4$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)