Question

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

Answer 1

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