Question

Which of the following definite intergrals have a non zero value

• A. ${\int }_0^{\pi /2}ln\left(cotx\right)dx$
• B. ${\int }_0^{2\pi }si{n}^{3}xdx$
• C. ${\int }_{1/e}^e\frac{dx}{x(lnx{)}^{1/3}}$
• D. None of these

Answer 1

Answer: (D)

None of these

Option A,
$I={\int }_0^{\pi /2}\ln \left(\cot x\right)dx$
$I={\int }_0^{\pi /2}ln\left(\tan x\right)dx$
$\Rightarrow 2I={\int }_0^{\pi /2}\ln \left(\tan x\cot x\right)dx$
$\Rightarrow 2I={\int }_0^{\pi /2}ln\left(1\right)dx$
$\Rightarrow I=0$
Option B,
$I={\int }_0^{2\pi }{\sin }^{3}xdx$
$I=\frac{1}{4}{\int }_0^{2\pi }(\sin 3x-3\sin x)dx$
$I=\frac{1}{4}(-\frac{1}{3}\cos 3x-3\cos x{)}_0^{2\pi }$
$\Rightarrow I=0$
Option C, ${\int }_{1/e}^e\frac{dx}{x(lnx{)}^{1/3}}$
Put $\ln x=t$
$\frac{1}{x}dx=dt$
$I={\int }_{-1}^1\frac{dt}{t^{1/3}}$
$I=\frac{2}{3}[t^{2/3}{]}_{-1}^1$