Integrate the rational function 1 (e^ x -1 ) [Hint: e^x = t] - Vidyadip

Question

Integrate the rational function $\frac{1}{\left(e^{x}-1\right)} [$Hint: $e^x = t]$

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Answer

Given function is, $\frac{1}{\left(e^{x}-1\right)}$
Let $e^x = t$
$e^x dx = dt$
$\int \frac{1}{\left(e^{x}-1\right)} d x=\int \frac{1}{t-1} \times \frac{d t}{t}=\int \frac{1}{t(t-1)} d t$
Let $\frac{1}{t(t-1)}=\frac{A}{t}+\frac{B}{t-1}$
$1 = A(t - 1) + Bt … (i)$
Substituting $t =1$ and $t = 0$ in equation $(i),$ we get,
$A = -1$ and $B = 1$
Therefore, $\frac{1}{t(t-1)}=\frac{-1}{t}+\frac{1}{t-1}$
$\int \frac{1}{t(t-1)} d t=\log \left|\frac{t-1}{t}\right|+C$
$= \log \left|\frac{e^{x}-1}{e^{x}}\right|+C$

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