Question

How do you integrate `int xe^(x^2)` using substitution?

Answer 1

Use substitution.

Explanation:

Let `u = x^2`. This make `du = 2x dx`

`int xe^(x^2) dx = 1/2 int underbrace((e^(x^2)))_(e^u) underbrace(2xdx)_(du)`

So we have

`int xe^(x^2) dx = 1/2 int (e^(x^2)) 2xdx`

`= 1/2 int e^u du`

`= 1/2e^u +C`

`= 1/2e^(x^2) +C`

(Check the answer by differentiating.)