Quick question I'm having trouble with (partial fraction integration)

[IntegrateMe]

Ok, so why do some answers contain an A/something + (Bx + C)/something and others contain just an A/something + B/something.

I hope someone knows what I'm talking about!

 

[Mark44]

Yes, I know what you're talking about. It depends whether the factors in the denominator are linear factors or irreducible quadratic factors.

For example, with 3/[(x - 1)(x² + 1)], the x - 1 factor in the denominator is a linear factor (x is to power 1) and x² + 1 is an irreducible quadratic. The decomposition would look like this:
3/[(x - 1)(x² + 1)] = A/(x - 1) + (Bx + C)/(x² + 1)

To reduce 3/[(x - 1)(x - 2)], both factors in the denominator are linear factors, so the decomposition would look like this:
3/[(x - 1)(x - 2)] = A/(x - 1) + B/(x - 2)

If you have repeated linear factors, such as 5/[(x - 1)², things are a little different and the decomposition would be this:
5/[(x - 1)² = A/(x - 1) + B/(x - 1)²

Hope that helps.

 

[IntegrateMe]

Thank's Mark, that makes complete sense. I was confused when the seemingly random x would tag onto an A or B term and i had no idea why. Thanks again for clarifying.