Question

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

• A. $3+2\pi$
• B. $4-\pi$
• C. $2+\pi$
• D. $2-\pi$

Answer 1

The correct option is B

$4-\pi$

${\int }_0^{\log 5}\frac{e^x\sqrt{e^x-1}}{e^x+3}dx$

Let $e^x+3=t$

$\Rightarrow e^{x}dx=dt$

Integration becomes ${\int }_4^8\frac{\sqrt{t-4}}{t}dt$

Let $t=4{\sec }^2\theta \Rightarrow dt=8{\sec }^2\theta \tan \theta d\theta$

Integration becomes ${\int }_0^{\frac{\pi }{4}}\frac{2\tan \theta \times 8{\sec }^2\theta \tan \theta }{4{\sec }^2\theta }d\theta ={\int }_0^{\frac{\pi }{4}}4({\sec }^2\theta -1)d\theta$

$=4(\tan \theta -\theta {)}_0^{\frac{\pi }{4}}$

$=4-\pi$