Question

How do you evaluate the integral `int x/(root3(x^2-1))`?

Answer 1

The integral equals `3/4(x^2 - 1)^(2/3) + C`

Explanation:

We will use u-substitution for this integral. Let `u = x^2 - 1`.

Then `du = 2xdx` and `dx= (du)/(2x)`. Call the integral `I`.

`I = int x/root(3)(u) * (du)/(2x)`

`I = 1/2 int 1/root(3)(u)`

`I = 1/2int u^(-1/3)`

`I = 1/2(3/2u^(2/3)) +C`

`I = 3/4u^(2/3) + C`

`I = 3/4(x^2 - 1)^(2/3) + C`

Hopefully this helps!