Integration Using Partial Fractions

[drmatth]

Homework Statement

Integrate (x^3 - 8x^2 - 1)/((x+3)(x^2-4x+5))

Homework Equations

This is an integration by partial fractions.

The Attempt at a Solution

http://www.wolframalpha.com/input/?i=integral+%28%28x^3-8x^2-1%29%2F%28%28x%2B3%29%28x^2-4x%2B5%29%29%29dx

I understand everything except where the integrand is rewritten after finding A, B, and C of the partial fraction decomposition. If anyone can help me understand how the integrand is rewritten that would be great. I just cannot make any sense out of it.

Thanks

Edit: If anyone is unfamiliar with WolframAlpha there is a "show steps" button in the top right corner of the problem statement, this is what I am referring to. It is about the 4th step down, rewrite the integrand.

 

[drmatth]

Here is a screenshot of the part I am asking about.

 

[Mark44]

They're starting with this part of the problem:
\int \frac{14 - 41x}{x^2 - 4x + 5}dx

What they're doing is manipulating things that that a substitution of u = x² - 4x + 5 (hence du = (2x - 4) dx) will work.

To get a numerator of -41/2*x + 82, which equals -(41/2)(2x - 4), they need to keep the numerator unchanged, so they are subtracting 68.

14 - 41x = -41x + 82 - 68 = -41/2(2x - 4) - 68

 

[drmatth]

Yep I see it. I noticed the differential so I thought it was a substitution but I just could not get my head around the manipulation. Thank you for breaking it down for me.