Question

The value of the integral ${\int }_0^{\log 5}\frac{e^x\sqrt{e^x-1}}{e^x+3}dx$ is

• A. $3+2\pi$
• B. $4-\pi$
• C. $2+\pi$
• D. none of these

Answer 1

Answer: (B)

$4-\pi$

Putting $e^x-1=t^2$ in the given integral, we have
${\int }_0^{\log 5}\frac{e^x\sqrt{e^x-1}}{e^x+3}dx=2{\int }_0^2\frac{t^2}{t^2+4}dt=2({\int }_0^{2}1dt-4{\int }_0^2\frac{dt}{t^2+4})$
$=2\left[{(t-2{\tan }^{-1}\left(\frac{t}{2}\right))}_0^2\right]$
$=2\left[\right(2-2\times \frac{\pi }{4}\left)\right]=4-\pi$