Integrate 2

Calculus Level 3

$\large \displaystyle\int_0^{\pi/2} \sin (2x)\log (\cot (x)) \ dx = \ ?$

Give your answer to three decimal places.

1 solution

Sandeep Rathod
Feb 25, 2015

$\displaystyle\int_0^{\pi/2} \sin 2x.\log \cot x \ dx$

$\displaystyle\int_0^{\pi/2} \sin 2x.\log \cos x \ dx - \displaystyle\int_0^{\pi/2} \sin 2x.\log \sin x \ dx$

$\displaystyle\int_0^{a} f(x) \ dx = \displaystyle\int_0^{a} f(a - x) \ dx$

$\displaystyle\int_0^{\pi/2} \sin 2x.\log \cos x \ dx = \displaystyle\int_0^{\pi/2} \sin 2x.\log \sin x \ dx$

Therefore answer is zero

Oh nice! That is a theorem, I didn't know. I used IBP rather.

Kartik Sharma - 6 years, 3 months ago

short and sweet! Nice

Bhargav Upadhyay - 6 years, 3 months ago