Question

How do you integrate by parts `(x*e^(4x)) dx`?

Answer 1

`int(x * e^(4x)dx) = 1/16 * e^(4x) * (4x-1) + c`

Explanation:

Remember that the formula that allows you to integrate a function by parts looks like this

`color(blue)(int(u * dv) = u * v - int(v * du))" "`, where

`u`, `v` - are two functions of `x`;
`du`, `dv` - their derivatives.

So, you need to identify `u` and `dv`, then calculate `du` and `v`.

If you take `u = x` and `dv = e^(4x)`, you will have

`u = x implies du = dx`

and

`dv = e^(4x) implies v = int(e^(4x)dx) = 1/4 * e^(4x)`

Your target integral will thus be

`int(x * e^(4x)dx) = x * 1/4 * e^(4x) - int(1/4 e^(4x) * dx)`

`int(x * e^(4x)dx) = 1/4 * x * e^(4x) - 1/4 * 1/4 * e^(4x) + c`

`int(x * e^(4x)dx) = 1/4e^(4x)(x - 1/4) + c`

`int(x * e^(4x)dx) = color(green)(1/16 * e^(4x) * (4x-1) + c)`