Question

How do you test the improper integral `int x^(-1/3)dx` from `[-1,0]` and evaluate if possible?

Answer 1

Use `int x^(-1/3) dx = 3/2x^(2/3)`.

Explanation:

`int_-1^0 x^(-1/3) dx = lim_(brarr0^-)int_-1^b x^(-1/3) dx`

`= lim_(brarr0^-) [3/2x^(2/3)]_-1^b`

`= lim_(brarr0^-) 3/2 b^(2/3) - 3/2(-1)^(2/3)`

`= 0 - 3/2 = -3/2`