Question

Which of the following definite integral(s) vanishes?

• A. $\underset{0}{\overset{\pi /2}{\int }}\ln \left(\cot x\right)dx$
• B. $\underset{0}{\overset{2\pi }{\int }}{\sin }^{3}xdx$
• C. $\underset{1/e}{\overset{e}{\int }}\frac{dx}{x(\ln x{)}^{1/3}}$
• D. $\underset{0}{\overset{\pi }{\int }}\sqrt{\frac{1+\cos 2x}{2}}dx$

Answer 1

Answer: (A), (B), (C)

$\underset{1/e}{\overset{e}{\int }}\frac{dx}{x(\ln x{)}^{1/3}}$

$\left(i\right)I=\underset{0}{\overset{\pi /2}{\int }}\ln \left(\cot x\right)dx\Rightarrow I=\underset{0}{\overset{\pi /2}{\int }}\ln \left(\tan x\right)dx\text{Adding both}\Rightarrow 2I=\underset{0}{\overset{\pi /2}{\int }}\ln (\cot x\cdot \tan x)dx\Rightarrow I=0\left(ii\right)I=\underset{0}{\overset{2\pi }{\int }}{\sin }^{3}xdx=-\underset{0}{\overset{2\pi }{\int }}{\sin }^{3}xdx\Rightarrow I=0$
$\left(iii\right)$ At $x=\frac{1}{t},$
$I=\underset{e}{\overset{1/e}{\int }}\frac{-\left(\frac{1}{t^2}\right)dt}{-\frac{1}{t}(\ln t{)}^{1/3}}=-\underset{1/e}{\overset{e}{\int }}\frac{dt}{t(\ln t{)}^{1/3}}\Rightarrow I=-I\Rightarrow I=0$
$\left(iv\right)\sqrt{\frac{1+\cos 2x}{2}}>0\Rightarrow \underset{0}{\overset{\pi }{\int }}\sqrt{\frac{1+\cos 2x}{2}}dx>0$