Integration: Partial Fractions

[Alw]

How does this work? All i really understood from class was that you would factor the integrand and then somehow A and B were involved, and you would use systems of equations to find A and B. What's the middle ground? Thanks in advance!

 

[Jameson]

Alright, so let's say you have to integrate the following expression.

(1)$$\int \frac{1}{x^2-1}dx$$

It should first be noticed that this doesn't follow any of the standard integration rules, like the natural log one for example and that another method should be employed. So to break this up into partial fractions you should factor the denominator and split the expression into two fractions like so.

(2)$$\frac{1}{x^2-1}= \frac{A}{x-1}+\frac{B}{x+1}$$

Now multiply through by (x-1)(x+1) to get this into something more workable. Doing this you'll get:

(3)$$1=A(x+1)+B(x-1)$$

You can use this expression and your previous one to do a system of equations, but another method is much simpler. Let x=-1 so that the A term will be zero and you can solve for B. Now let x=1 so that the B term will be zero and you can solve for A.

I'll let you finish this one, but I hope the concept explanation helps!

Jameson