Question

If u represents integral of ((cos(x))/(sin(x)+cos(x)))^2 from 0 to pi /4 and
v represents integral of ((sin(x)+cos(x))/(cos(x)))^2 from 0 to pi/4 then find v/u.

Answer 1

$v=\int\limits_0^{\frac{\pi}{4}}~\left(\dfrac{\sin\,x+\cos\,x}{\cos\,x}\right)^2\,dx$

$~=\int\limits_0^{\frac{\pi}{4}}~\left(\dfrac{\sin\,(\frac{\pi}{4}-x)+\cos\,(\frac{\pi}{4}-x)}{\cos\,(\frac{\pi}{4}-x)}\right)^2\,dx$

$~=\int\limits_0^{\frac{\pi}{4}}~\left(\dfrac{\frac{1}{\sqrt{2}}\cos\,x-\frac{1}{\sqrt{2}}\sin\,x+\frac{1}{\sqrt{2}}\cos\,x+\frac{1}{\sqrt{2}}\sin\,x}{\frac{1}{\sqrt{2}}\cos\,x+\frac{1}{\sqrt{2}}\sin\,x}\right)^2\,dx$

$~=\int\limits_0^{\frac{\pi}{4}}~\left(\dfrac{\sqrt{2}\cos\,x}{\frac{1}{\sqrt{2}}(\cos\,x+\sin\,x)}\right)^2\,dx$

$~=4~\int\limits_0^{\frac{\pi}{4}}~\left(\dfrac{\cos\,x}{\cos\,x+\sin\,x}\right)^2\,dx$

$~=4u$

$\implies~\boxed{\frac{v}{u}=4}$