Question

$\int^{\pi/2}_{0} \log\left(\frac{cos x}{sin x}\right) dx$ is equal to

• A. $\pi$/2
• B. $\pi$/4
• C. $\pi$
• D. 2$\pi$
• E. 0

Answer 1

Answer: (E)

Let $I=\int_{0}^{\pi / 2} \log \left(\frac{\cos x}{\sin x}\right) d x=\int_{0}^{\pi / 2} \log (\cot x) d x \, ...(i)$
Then, $I=\int_{0}^{\pi / 2} \log \cot \left(\frac{\pi}{2}-x\right) d x$
$[\because \int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x]\, ....(ii) $
$=\int_{0}^{\pi / 2} \log \tan \,x \,d x$
On adding Eqs. (i) and (ii), we get
$2 I=\int_{0}^{\pi / 2}(\log \cot x+\log \tan x) d x$
$\Rightarrow 2 l=\int_{0}^{\pi / 2} 0 d x=0 $
$ \Rightarrow l=0$