Question

\[\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =\]

Options

• \[\frac{e^x}{x + 4} + C\]

• \[\frac{e^x}{x + 3} + C\]

• \[\frac{1}{\left( x + 4 \right)^2} + C\]

• \[\frac{e^x}{\left( x + 4 \right)^2} + C\]

Answer 1

\[\frac{e^x}{x + 4} + C\]

 

\[\text{Let }I = \int\frac{\left( x + 3 \right)}{\left( x + 4 \right)^2} e^x dx\]
\[ \Rightarrow \int\left[ \frac{x + 4 - 1}{\left( x + 4 \right)^2} \right] e^x dx\]
\[ \Rightarrow \int\left[ \frac{1}{\left( x + 4 \right)} - \frac{1}{\left( x + 4 \right)^2} \right] e^x dx\]
\[\text{As, we know that }\int e^x \left\{ f\left( x \right) + f'\left( x \right) \right\} dx = e^x f\left( x \right) + C\]
\[ \therefore I = \frac{e^x}{x + 4} + C\]