Evaluate the following functions : e^x+1 e^x-1 d x - Vidyadip

Question

Evaluate the following functions : $\int \frac{e^x+1}{e^x-1} \cdot d x$

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Answer

$
\begin{aligned}
& I=\int \frac{e^x-1+2}{e^x-1} \cdot d x \\
& =\int\left(\frac{e^x-1}{e^x-1}+\frac{2}{e^x-1}\right) \cdot d x \\
& =\int\left(1+\frac{2}{e^x-1}\right) \cdot d x \\
& =\int d x+\int \frac{2}{e^x\left(1-e^{-x}\right)} \cdot d x \\
& =\int 1 d x+2 \int \frac{e^{-x}}{1-e^{-x}} \cdot d x \\
& \text { put }\left(1-e^{-x}\right)=t \\
& \text { Differentiate w.r.t. } x \\
& -\left(e^{-x}\right)(-1) \cdot d x=1 d t \\
& e^{-x} \cdot d x=1 d t \\
& \mathrm{I}=\int 1 d x+2 \int \frac{1}{t} \cdot d t \\
& =x+2 \cdot \log (t)+c \\
& =x+2 \log \left(1-e^{-x}\right)+c \\
& \therefore \quad \int \frac{e^x+1}{e^x-1} \cdot d x=x+2 \log \left(1-e^{-x}\right)+c \\
\end{aligned}
$

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