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Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
If $\int_{\log 2}^x \frac{ du }{\left( e ^{ u }-1\right)^{1 / 2}}=\frac{\pi}{6}$, then $e ^x=$
  • A
    1
  • B
    2
  • ✓
    4
  • D
    -1
✓

Answer

Correct option: C.
4
(C)
$\int_{\log 2}^x \frac{ du }{\left( e ^{ u }-1\right)^{1 / 2}}=\frac{\pi}{6} \Rightarrow \int_{\log 2}^x \frac{ e ^{ u }}{ e ^{ u }\left( e ^{ u }-1\right)^{1 / 2}} du =\frac{\pi}{6}$
Put $e^u-1=t^2 \Rightarrow e^u d u=2 t d t$
When $u =\log 2, t =1$
and when $u =x, t =\sqrt{ e ^x-1}$
$\therefore \quad \int_1^{\sqrt{ e ^x-1}} \frac{2}{1+ t ^2} dt =\frac{\pi}{6}$
$\Rightarrow 2\left[\tan ^{-1} t \right]_1^{\sqrt{ e ^x-1}}=\frac{\pi}{6}$
$\Rightarrow \tan ^{-1}\left(\sqrt{ e ^x-1}\right)-\frac{\pi}{4}=\frac{\pi}{12}$
$\Rightarrow \sqrt{ e ^x-1}=\tan \frac{\pi}{3} \Rightarrow \sqrt{ e ^x-1}=\sqrt{3} \Rightarrow e ^x=4$

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