Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration2 Marks
Question
Integrate the following functions w.r.t. x:
$\frac{e^{2 x}+1}{e^{2 x}-1}$
✓
Answer
Let $I=\int \frac{e^{2 x}+1}{e^{2 x}-1} d x=\int \frac{\left(\frac{e^{2 x}+1}{e^x}\right)}{\left(\frac{e^{2 x}-1}{e^x}\right)} d x$
$=\int\left(\frac{e^x+e^{-x}}{e^x-e^{-x}}\right) d x$
$=\int \frac{\frac{d}{d x}\left(e^x-e^{-x}\right)}{e^x-e^{-x}} d x$
$=\log \left|e^x-e^{-x}\right|+c \quad \cdots\left[\because \int \frac{f^{\prime}(x)}{f(x)} d x=\log |f(x)|+c\right]$
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