Question

Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.)
(11)/(x^3-125) dx

Answer 1

using partial fractions, you have

11 * 1/[(x-5)(x^2+5x+25)]
= 11/75 * [1/(x-5) - (x+10)/(x^2+5x+25)]

Doesn't look much better, does it? Well, the 1/(x-5) integrates easily enough. The other term has to be worked into something more standard.

x^2+5x+25 = (x + 5/2)^2 + 75/4

Now you have

(x+10)/[(x + 5/2)^2 + 75/4]
= x/(x^2+5x+25) + 10/[(x + 5/2)^2 + 75/4]

The x/(x^2+5x+25) integrates easily enough, and you are left with the other part. Letting

x + 5/2 = 5?3/2 tan?
(x + 5/2)^2 + 75/4 = 75/4 sec^2?
dx = 5?3/2 sec^2? d?
and the integrand becomes just 4?3 d?

Putting all that together you get the result shown here:

http://www.wolframalpha.com/input/?i=%E2%88%AB11%2F(x%5E3-125)+dx