Question

Let \[ f(x)=\int \frac{x\sqrt{x}}{(1+x)^2}\,dx \qquad (x\ge0) \] Then $f(3)-f(1)$ is equal to:

• A. \(\dfrac{-1}{12}+\dfrac{1}{2}+\dfrac{\sqrt{3}}{4}\)
• B. \(\dfrac{1}{12}+\dfrac{1}{2}-\dfrac{\sqrt{3}}{4}\)
• C. \(\dfrac{-1}{6}+\dfrac{1}{2}+\dfrac{\sqrt{3}}{4}\)
• D. \(\dfrac{1}{6}+\dfrac{1}{2}-\dfrac{\sqrt{3}}{4}\)

Hint:

For integrals involving \(\sqrt{x}\), substitution \(x=t^2\) usually simplifies the expression.

Answer 1

The Correct Option is A