Question

Let \[ f(x)=\int \frac{\sqrt{x}}{(1+x)^2}\,dx \quad (x\geq 0) \] Then \[ f(3)-f(1) \] is equal to:

• A. $ -\frac{\pi}{12} + \frac{1}{2} + \frac{\sqrt{3}}{4} $
• B. $ \frac{\pi}{12} + \frac{1}{2} - \frac{\sqrt{3}}{4} $
• C. $ -\frac{\pi}{6} + \frac{1}{2} + \frac{\sqrt{3}}{4} $
• D. $ \frac{\pi}{6} + \frac{1}{2} - \frac{\sqrt{3}}{4} $

Hint:

When you see $ \sqrt{x} $ and $ (1+x) $ in the same integral, the substitution $ x = \tan^2 \theta $ is usually the fastest way to collapse the denominator using Pythagorean identities.

Answer 1

The Correct Option is B