Question

How do you find the antiderivative of `x^2 e^(2x)`?

Answer 1

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

Explanation:

Well, This can't be done simply.

We have to Perform Integration By Parts.

The Rule States That,

`int uvdx = uintvdx - int (u'intv)dx`

Where `u` and `v` are functions of `x` and `u' = d/dx(u)`.

Now, We have,

`intx^2e^(2x)dx`

`= x^2inte^2x - int(d/dx(x^2)inte^(2x))dx`

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

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

We have to Integrate By Parts Once More.

So, From (i),

`1/2x^2e^(2x) - x inte^(2x) + int(d/dx(x)inte^(2x))dx`

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

`= 1/2x^2e^(2x) - 1/2xe^(2x) + 1/4e^(2x) + C` [Yeah, Don't EVER Forget this guy here`rarr` `C`]

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

Hope this helps.