Solve this

Question:

Let $f(x)=\int \frac{\sqrt{x}}{(1+x)^{2}} d x(x \geq 0) .$ Then $f(3)-f(1)$ 

is equal to :

  1. $-\frac{\pi}{6}+\frac{1}{2}+\frac{\sqrt{3}}{4}$

  2. $\frac{\pi}{6}+\frac{1}{2}-\frac{\sqrt{3}}{4}$

     

  3. $-\frac{\pi}{12}+\frac{1}{2}+\frac{\sqrt{3}}{4}$

  4. $\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}$


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]

$=-\left(\frac{\sqrt{3}}{4}-\frac{1}{2}\right)+\frac{\pi}{3}-\frac{\pi}{4}$

$=\frac{1}{2}-\frac{\sqrt{3}}{4}+\frac{\pi}{12}$

 

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