Tricky integral using Partial Fractions

[thomas49th]

Homework Statement

The question asks me to express the integrand in partial fractions to evaluate the integral

\int \frac{13x-4}{6x^{2} -x -2} dx

Homework Equations

The Attempt at a Solution

Well 6x² -x - 2 doesn't factorise (or I can't see it factorised).

So I tried doing long polynomial division and I got something that looked nothing like what the answer is. How should I begin to tackle this problem?

Thanks
Thomas

 

[Gregg]

This thread should be helpful

https://www.physicsforums.com/showthread.php?t=337025

 

[Count Iblis]

6x² -x - 2 = 1/6 (6x + 3)(6x - 4)

 

[Count Iblis]

And as Gregg suggested you can remove the term linear in x (i.e. 13 x) term from the numerator before you start. That will save you work on the partial fraction expansion:

You can write: 13 = 12 * (13/12)

The numerator can thus be written as:

13/12 (12 x - 48/13) =

13/12 (12 x - 1 + 1 - 48/13) =

13/12 (derivative of denominator -35/13) =

13/12 derivative of denominator - 35/12

So, the integral will be 13/12 Log[6 x^2 - x -2] plus the integral of

-35/12 1/(6 x^2 - x -2)