Question

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

Answer 1

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