Question:
Let $f(x)=\int \frac{\sqrt{x}}{(1+x)^{2}} d x(x \geq 0) .$ Then $f(3)-f(1)$
is equal to :
Correct Option: , 4
Solution:
$f(x)=\int_{i}^{3} \frac{\sqrt{x} d x}{(1+x)^{2}}=\int_{1}^{\sqrt{3}} \frac{t .2 t d t}{\left(1+t^{2}\right)^{2}} \quad($ put $\sqrt{x}=t)$
$=\left(-\frac{t}{1+t^{2}}\right)_{1}^{\sqrt{3}}+\left(\tan ^{-1} t\right)_{1}^{\sqrt{3}} \quad$ [Appling by parts]
$=-\left(\frac{\sqrt{3}}{4}-\frac{1}{2}\right)+\frac{\pi}{3}-\frac{\pi}{4}$
$=\frac{1}{2}-\frac{\sqrt{3}}{4}+\frac{\pi}{12}$