Question

Question #0f959

Answer 1

`4/3(x^2+x+1)^(3/2)+C`

Explanation:

We have:

`int(4x+2)(x^2+x+1)^(1/2)dx`

You may not realize it yet, but this is set up quite well for substitution. We will use the substitution `u=x^2+x+1`. Note that this implies that `du=(2x+1)dx`.

Factor a `2` from the `(4x+2)` term:

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

Notice both our `u` and `du` terms are present:

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

`=2intu^(1/2)du`

Using the rule `intu^ndu=u^(n+1)/(n+1)+C`:

`=2(u^(3/2)/(3/2))+C`

`=4/3u^(3/2)+C`

`=4/3(x^2+x+1)^(3/2)+C`