Verified answer

I was going to integrate it by parts, but then I realized it'd just be easier to look at the answer choices and take the derivative of choices that look like possible answers.

The answer is C.

(1/4)(3e^x + 1)^4/3 + C

Proof:

Taking the derivative, first using the power rule and then the chain rule.

(1/4)[4/3][3e^x](3e^x + 1)^[4/3 - 1]

(1/4)[4e^x](3e^x + 1)^[1/3]

e^x(3e^x + 1)^(1/3)

And to do it the longer but more proper way, here's how you take the integral:

∫ e^x(3e^x + 1)^1/3 dx

We’re going to use integration by parts.

u = (3e^x + 1)^1/3 du = e^x(3e^x + 1)^-2/3 dx

dv = e^x v = e^x

∫u dv = uv - ∫v du

∫ e^x(3e^x + 1)^1/3 dx = [(3e^x + 1)^1/3][e^x] - ∫[e^x][e^x(3e^x + 1)^-2/3] dx =

You need to use substitution for our ∫vdu.

w = (3e^x – 1)

dw = 3e^x dx

Also note: e^x = (w+1)/3

Therefore that integral becomes:

∫[e^x][e^x(3e^x + 1)^-2/3] dx = 1/3∫[(w+1)/3(w)^-2/3] dw =

1/3∫[(w+1)/[3w^2/3]] dw = 1/3∫[[1/3 w^(1/3) + 1/3 w^-2/3]] dw =

1/9∫[[w^(1/3) + w^-2/3]] dw = 1/9 [ 3/4 w^(4/3) + 3w^(1/3)] + C =

1/12 w^(4/3) + 1/3 w^(1/3) + C

Plug w back in and you get:

1/12 (3e^x – 1)^(4/3) + 1/3 (3e^x – 1)^(1/3) + C

So our integral becomes:

[(3e^x + 1)^1/3][e^x] - 1/12 (3e^x – 1)^(4/3) + 1/3 (3e^x – 1)^(1/3) + C

And eventually

(1/4)(3e^x + 1)^4/3 + C