Question

Write a value of $\int e^{ax}\left\{af\left(x\right)+f\text{'}\left(x\right)\right\}dx$.

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int e^{ax}\left[af\left(x\right)+f\text{'}\left(x\right)\right]dx \\ \mathrm{Let}{e}^{ax}.f\left(x\right)=t \\ \Rightarrow \left[e^{ax}.af\left(x\right)+e^{ax}.f\text{'}\left(x\right)\right]dx=dt \\ \therefore I=\int dt \\ =t+C \\ =e^{ax}.f\left(x\right)+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathit{\because }\mathit{t}\mathit{=}{\mathit{e}}^{\mathit{a}\mathit{x}}\mathit{.}\mathit{f}\left(\mathit{x}\right)\right) \end{gathered}$$