Question

$\int \frac{e^x}{e^{2x}+5e^x+6}dx$

Answer 1

$$\begin{gathered} \int \frac{e^{x}dx}{e^{2x}+5e^x+6} \\ \mathrm{let}{e}^x=t \\ \Rightarrow e^{x}dx=dt \\ \mathrm{Now},\int \frac{e^{x}dx}{e^{2x}+5e^x+6} \\ =\int \frac{dt}{t^2+5t+6} \\ =\int \frac{dt}{t^2+5t+{\left({\displaystyle \frac{5}{2}}\right)}^2-{\left({\displaystyle \frac{5}{2}}\right)}^2+6} \\ =\int \frac{dt}{{\left(t+{\displaystyle \frac{5}{2}}\right)}^2-{\displaystyle \frac{25}{4}}+6} \\ =\int \frac{dt}{{\left(t+{\displaystyle \frac{5}{2}}\right)}^2-{\displaystyle \frac{25+24}{4}}} \\ =\int \frac{dt}{{\left(t+{\displaystyle \frac{5}{2}}\right)}^2-{\left({\displaystyle \frac{1}{2}}\right)}^2} \\ =\frac{1}{2\times {\displaystyle \frac{1}{2}}}\log \left|\frac{t+{\displaystyle \frac{5}{2}}-{\displaystyle \frac{1}{2}}}{t+{\displaystyle \frac{5}{2}}+{\displaystyle \frac{1}{2}}}\right|+C \\ =\log \left|\frac{t+2}{t+3}\right|+C \\ =\log \left|\frac{e^x+2}{e^x+3}\right|+C \end{gathered}$$