Question

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

Answer 1

\[\int\frac{e^x dx}{e^{2x} + 5 e^x + 6}\]
\[\text{ let } e^x = t\]
\[ \Rightarrow e^x \text{ dx }= dt\]
\[Now, \int\frac{e^x dx}{e^{2x} + 5 e^x + 6}\]
\[ = \int\frac{dt}{t^2 + 5t + 6}\]
\[ = \int\frac{dt}{t^2 + 5t + \left( \frac{5}{2} \right)^2 - \left( \frac{5}{2} \right)^2 + 6}\]
\[ = \int\frac{dt}{\left( t + \frac{5}{2} \right)^2 - \frac{25}{4} + 6}\]
\[ = \int\frac{dt}{\left( t + \frac{5}{2} \right)^2 - \frac{25 + 24}{4}}\]
\[ = \int\frac{dt}{\left( t + \frac{5}{2} \right)^2 - \left( \frac{1}{2} \right)^2}\]
\[ = \frac{1}{2 \times \frac{1}{2}} \log \left| \frac{t + \frac{5}{2} - \frac{1}{2}}{t + \frac{5}{2} + \frac{1}{2}} \right| + C\]
\[ = \text{ log }\left| \frac{t + 2}{t + 3} \right| + C\]
\[ = \text{ log}  \left| \frac{e^x + 2}{e^x + 3} \right| + C\]