Question

$\underset{\ln x-\ln 2}{\overset{\ln \pi }{\int }}\frac{e^x}{1-\cos \left(\frac{2}{3}{e}^x\right)}dx$ is equal to

• A. $\sqrt{3}$
• B. $-\sqrt{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. $-\frac{1}{\sqrt{3}}$

Answer 1

Answer: (A)

$\underset{\ln \pi -\ln 2}{\overset{\ln \pi }{\int }}\frac{e^x}{1-\cos \left(\frac{2}{3}{e}^x\right)}dx=\frac{3}{2}\underset{\pi /3}{\overset{2\pi /3}{\int }}\frac{dt}{1-\cot }=\frac{3}{2}\underset{\pi /3}{\overset{2\pi /3}{\int }}\frac{dt}{1-\left(1-2{\sin }^{2}t\right)}$
$=\frac{3}{2.2}\underset{\pi /3}{\overset{2\pi /3}{\int }}{\mathrm{cosec}}^{2}tdt=\frac{3}{4}[-\cot t{]}_{\pi /3}^{2\pi /3}$
$=-\frac{3}{4}\left[-\sqrt{3}-\frac{1}{\sqrt{3}}\right]=\frac{3}{4}\left(\frac{4}{\sqrt{3}}\right)=\sqrt{3}$