Question

Evaluate \[ \int_{0}^{\pi/2} \frac{x\sin 2x}{1+4\cos^2 2x}\,dx. \]

• A. \[ \frac{\pi}{8}\tan^{-1}2 \]
• B. \[ \frac{\pi}{4}\tan^{-1}2 \]
• C. \[ \frac{\pi}{2}\tan^{-1}2 \]
• D. \[ \frac{\pi}{8}\tan^{-1}4 \]

Hint:

For definite integrals involving \(x\), always check whether \[ f(a-x)=f(x). \] If true, use \[ \int_0^a x f(x)\,dx = \frac{a}{2}\int_0^a f(x)\,dx, \] which greatly simplifies the calculation.

Answer 1

The Correct Option is A