Integration, please help!

Question

Evaluate  

Options:

1. 
2. -ln|1 - x³| + C
3. sin^-1(x³) + C
4. 

Answer 1

U = 1 - x^3

dU = -3x^2 dx

integral ( 3x^2 dx / (1 - x^3) ) =

integral ( -du/U) = - integral ( du/U) = -ln|U| = -ln| 1 - x^3| + c

option #2

Answer 2

This is accomplished by doing a "u" substitution. You can see that, because the derivative of what's inside the complex function, x³, is 3x² and that is what is on top.

So let u = 1 - x³ then du/dx = -3x² or dx = du/-3z²

The integral becomes:

Simplifying:

-∫u^-1/2du = -2u^1/2 = -2(1 - x³)^1/2 + C (answer 4)