Question

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

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{a}{b+c{e}^x}dx \\ \mathrm{Dividing}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}{e}^x \\ \Rightarrow I=\int \frac{a{e}^{-x}}{b{e}^{-x}+c}dx \\ \mathrm{Putting}{e}^{-x}=t \\ \Rightarrow -e^{-x}=\frac{dt}{dx} \\ \Rightarrow e^{-x}dx=-dt \\ \therefore I=\int \frac{-a}{bt+c}dt \\ =\frac{-a}{b}\ln \left|bt+c\right|+C\left[\because \int \frac{1}{ax+b}dx=\frac{1}{a}\ln \left|ax+b\right|+\mathrm{C}\right] \\ =\frac{-a}{b}\ln \left|b{e}^{-x}+c\right|+C\left[\because t=e^{-x}+C\right] \end{gathered}$$