Question

Integrate `int xe^(2x)dx` ?

Answer 1

This will require a single application of integration by parts.

We let `dv = e^(2x)dx` and `u = x`. Then `v= 1/2e^(2x)` and `du = dx`.

By integration by parts we have

`int u dv = uv - int v du`

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

`int xe^(2x) dx = 1/2xe^(2x) - 1/4e^(2x) + C`

Hopefully this helps!

Answer 2

The answer is `=(2x-1)e^(2x)/4+C`

Explanation:

Perform an integration by parts

`intuv'dx=uv-intu'vdx`

Here,

The integral is

`I=intxe^(2x)dx`

`u=x`, `=>`, `u'=1`

`v'=e^(2x)`, `=>`, `v=e^(2x)/2`

Therefore, the integral is

`I=(xe^(2x))/2-1/2inte^(2x)dx`

`=(xe^(2x))/2-1/4e^(2x)+C`

`=(2x-1)e^(2x)/4+C`