Indefinite Integration of a Rational Expression

[communitycoll]

Homework Statement

<Indefinite integral sign here>[r^2 -2r] / [r^3 - 3r^2 + 1]dr
or the second example in the "Substitution" section here:
http://people.clarkson.edu/~sfulton/ma132/parfrac.pdf

Homework Equations

nada.

The Attempt at a Solution

Nothing to really attempt, I just don't get what they do with the numerator (i.e., how it turns into a 1).

 

[tiny-tim]

hi communitycoll!
(have an integral: ∫ and try using the X² button just above the Reply box )

… I just don't get what they do with the numerator (i.e., how it turns into a 1).

ah, nooo, the numerator isn't r² - 2r, it's (r² - 2r)dr !

(and of course that's 1 times dw )

the trick in substitution is that you always have to substitute the "d" part also!

 

[eumyang]

If you read that attachment carefully, you would have seen this:

Solution: Here we notice that the numerator is the derivative of the denominator (to within a constant factor).

So they let
w = r³ - 3r² + 1,
so
dw = 3r² - 6r dr,
which is the same as
dw = 3(r² - 2r) dr.

Do you see it now?

 

[communitycoll]

Okay then. Yeah, I understand now. Thanks.