Question

Let \[ f(x)=\int \frac{\sqrt{x}}{(1+x)^2}\,dx, \quad (x \ge 0) \]

Then, find the value of: \[ f(3)-f(1) \] 

• 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:

Whenever an integral contains expressions like $\sqrt{x}$ together with $(1+x)$, the substitution \[ x=\tan^2\theta \] is extremely effective because it converts \[ 1+\tan^2\theta \] directly into \[ \sec^2\theta. \]

Answer 1

The Correct Option is B