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Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{a}{b+c e^{x}} d x$

Solution:

First of all take $e^{x}$ common from denominator so we get

$\Rightarrow \int \frac{a}{e^{x}\left(\frac{b}{e^{x}}+c\right)} \cdot d x$

$\Rightarrow \int \frac{a \cdot e^{-x}}{b e^{-x}+c} d x$

Assume be $^{-x}+c=t$

$d\left(b e^{-x}+c\right)=d t$

$\Rightarrow-\mathrm{be}^{-\mathrm{x}} \mathrm{dx}=\mathrm{dt}$

$\Rightarrow \mathrm{e}^{-\mathrm{x}} \mathrm{dx}=\frac{-\mathrm{dt}}{\mathrm{b}}$

Substituting $t$ and dt we get

$\Rightarrow \int \frac{-a d t}{b t}$

$\Rightarrow \frac{-a}{b} \ln |t|+c$

But $t=\left(b e^{-x}+c\right)$

$\Rightarrow \frac{-a}{b} \ln \left|b e^{-x}+c\right|+c$

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