Question

How do you find the antiderivative of `int x^2/sqrt(1-x^3)dx`?

Answer 1

`-2/3sqrt(1-x^3)+C`

Explanation:

Rewriting the function:

`I=intx^2/sqrt(1-x^3)dx=intx^2(1-x^3)^(-1/2)dx`

To deal with the `-1//2` power, let `u=1-x^3`. This implies that `du=-3x^2color(white).dx`. Rewriting the integral:

`I=-1/3int(1-x^3)^(-1/2)(-3x^2color(white).dx)=-1/3intu^(-1/2)color(white).du`

Now use the rule `intu^ncolor(white).du=u^(n+1)/(n+1)`:

`I=-1/3(u^(1/2)/(1/2))=-2/3sqrtu=-2/3sqrt(1-x^3)+C`