question_answer

Evaluate \[\int{{{e}^{ax}}\{af(x)+f'(x)\}\,dx.}\]

Answer:

Let\[l=\int{{{e}^{^{ax}}}\{af\,(x)+f'(x)\}\,dx}\]Put \[ax=t\] \[\Rightarrow \]   \[a\,dx=dt\] \[\Rightarrow \]   \[dx=\frac{1}{a}dt\] \[\therefore \]      \[\left. l=\frac{1}{a}\int{{{e}^{t}}}\left\{ af\left( \frac{t}{a} \right)+f'\left( \frac{t}{a} \right)dt \right\} \right\}\] \[=\frac{1}{a}{{e}^{t}}\,\,af\left( \frac{t}{a} \right)+C\] \[={{e}^{t}}\,\,(f(t))+C\] \[={{e}^{ax}}\,\,f\,(x)+C\]