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