Question

How do you integrate `int x^2e^(2x)` by parts?

Answer 1

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

Explanation:

Take `x^2` as finite part, so in the integration by parts the degree of `x` decreases:

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

We can now solve the resulting integral by parts again:

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

and we can now solve the last integral directly:

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

Putting it all together:

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