Question

Question #2efd1

Answer 1

U-substitution, see explanation

Explanation:

I will presume that you meant to write `int e^(x^3)x^2dx`. If you did not mean this, then the solution below will not help you.

If so, we can solve this problem via u-substitution. In u-substitution, we essentially invert the chain rule; we recognize an integrand as being an instance of `f'(g(x))*g'(x)`, and define `g(x)=u, g'(x)=du,`, allowing us to instead describe the integrand as `f'(u)du`.

In this case, if we take `x^3` to be our `u`, then `e^(x^3) = e^u, du = 3x^2`. Note that we instead have `x^2` in our integrand; this must mean that the initial function had a `1/3` before differentiation. Then we have:

`int e^(x^3)x^2dx = int 1/3e^udu`

By the definition of the derivative and integral of `e^x`, this becomes:

`= 1/3 e^u +c`

Putting the equation back in terms of x...

`= 1/3 e^(x^3)+c`