Question

Evaluate the integral

${\int }_0^{\pi /2}$ $\sin 2x\log \left(\tan x\right)dx$

• A. $0$
• B. $-1/2$
• C. $1/3$
• D. $1/4$

Answer 1

Answer: (A)

$0$

${\int }_0^{\frac{\pi }{2}}\sin 2x\log \left(\tan x\right)dx$ ---(1)

${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$

$I={\int }_0^{\frac{\pi }{2}}+\sin 2x\log \left(cotx\right)dx$

$={\int }_0^{\frac{\pi }{2}}\sin 2x\log \left(\frac{1}{\tan x}\right)dx$

$=-{\int }_0^{\frac{\pi }{2}}\sin 2x\log \left(\tan x\right)dx$ ---(2)

Adding $1$ and $2$, we get

$2I=0$

Thus $I=0$