The correct option is D π12+12−√34
Given:
f(x)=∫√x(1+x)2dx
Put x=tan2θ
⇒dx=2tanθsec2θdθ
∴f(θ)=∫2tan2θ.sec2θsec4θd θ
⇒f(θ)=∫2sin2θ d θ
⇒f(θ)=θ−sin 2θ2+C
⇒f(θ)=θ−12×2tanθ1+tan2θ+C
⇒f(θ)=θ−tanθ1+tan2θ+C
∴f(x)=tan−1√x−√x1+x+C
Now,
f(3)−f(1)=tan−1(√3)−√31+3−tan−1(1)+12
∴f(3)−f(1)=π12+12−√34