1. Tardigrade
  2. Question
  3. Mathematics
  4. If ∫ limits log 2x (1/√ex-1) d x=(π/6), then the value of x is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If $\int\limits_{\log 2}^{x} \frac{1}{\sqrt{e^{x}-1}} d x=\frac{\pi}{6}$, then the value of $x$ is

Bihar CECEBihar CECE 2013

Solution:

Let $l=\int\limits_{\log 2}^{x} \frac{1}{\sqrt{e^{x}-1}} d x$
Put $e^{x}-1=t^{2}$
$\Rightarrow e^{x} d x=2 t\, d t$
$\therefore l=2 \int\limits_{1}^{\sqrt{e^{x}-1}} \frac{1}{1+t^{2}} d t$
$=2\left[\tan ^{-1} \sqrt{e^{x}-1}-\tan ^{-1} 1\right]$
$=2 \tan ^{-1} \sqrt{e^{x}-1}-\frac{\pi}{4}$
But $\int\limits_{\log _{0}^{2}}^{x} \frac{1}{\sqrt{e^{x}-1}} d x=\frac{\pi}{6}$
$=2 \tan ^{-1} \sqrt{e^{x}-1}$
$=\frac{2 \pi}{3}$
$\Rightarrow \sqrt{e^{x}-1}=\tan \frac{\pi}{3}$
$=\sqrt{3}$
$=e^{x}-1$
$=3$
$\Rightarrow e^{x}=4$
$\therefore x=\log 4$