Integrate the rational functions- -1- e x-1 dx

Let I=∫1/e^x 1dx On multiplying numerator and denominator by e^ x, we get I=∫e^ x/1 e^ xdx Put 1 e^ x=t ⇒ e^ x( 1)=dt/dx⇒ dx =dt/e^ x⇒ e^ xdx =dt ∴ I =∫dt/t=lo ...

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Standard XII

Mathematics

Standard Formulae - 2

Integrate the...

Question

Integrate the rational functions.
$\int \frac{1}{e^{x} - 1} d x$

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Solution

Let $I = \int \frac{1}{e^{x} - 1} d x$
On multiplying numerator and denominator by $e^{- x}$ , we get
$I = \int \frac{e^{- x}}{1 - e^{- x}} d x$
Put $1 - e^{- x} = t \Rightarrow - e^{- x} (- 1) = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{e^{- x}} \Rightarrow e^{- x} d x = d t$
$∴ I = \int \frac{d t}{t} = l o g | t | + C = l o g | 1 - e^{- x} | + C = l o g ∣ ∣ \frac{e^{x} - 1}{e^{x}} ∣ ∣ + C$

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