Write a value of `inte^(a x)\ {a\ f\ (x)+f^(prime)(x)}\ dx`

Explanation

Solution:

`I=\int e^{a x}{af}(x)+f^{\prime}(x)] d x`

`I=a \int e^a x f(x) d x+\int e^a x f^{\prime}(x) d x`

Let `u=e^{a x} \Rightarrow d u=a e^{a x} d x`

`d v=f^{\prime}(x) \Rightarrow v=f(x)`

`\int u \cdot v d x=u \int v d x-\int[\int v d x \cdot \frac{d u}{d x} \cdot d x\] \ldots` . . . by parts formula.

`I=a \int e^{a x} f(x) d x+e^{a x} f(x)-a \int e^{a x} f(x) d x+c`

`\therefore I=e^ ax f(x)+c`