Question

The value of the integral $\underset{0}{\overset{\frac{\pi }{2}}{\int }}log\left(tanx\right)dx=$

• A. 0
• B. 1
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (A)

Let I = $\underset{0}{\overset{\frac{\pi }{2}}{\int }}$ log (tan x)dx ...(1)
Then, I = $\underset{0}{\overset{\frac{\pi }{2}}{\int }}log\left[tan\right(\frac{\pi }{2}-x\left)\right]dx$
$[\because \underset{0}{\overset{a}{\int }}f(x)dx=\underset{0}{\overset{a}{\int }}f(a-x)dx]$
$\Rightarrow I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}log(cotx)dx$
$\Rightarrow I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}log\left(\frac{1}{tanx}\right)dx$
$\Rightarrow I=\underset{0}{\overset{\frac{\pi }{2}}{\int }}log(tanx{)}^{-1}dx=-\underset{0}{\overset{\frac{\pi }{2}}{\int }}log(tanx)dx$
$\Rightarrow$ I = -I $\Rightarrow$ 2I = 0 $\Rightarrow I=0$
[Using eq. (1)]