Calc 2 How do I evaluate this Integral?

Question

Integral ( (x^2 + 2x - 1) / (x^3 - 1) dx)
I know I need to do partial fractions, so I get the bottom to be (x - 1)(x^2 + x + 1),
but I'm really bad at setting these up...
Would it be:
A/(x - 1) + (Bx + C)/(x^2 + x + 1)? How do you know when to use Bx + C? How do you know when to just use Bx, and how do yo ukonw when you need to split up up further (like I know you do it when it is an x^2..?) 

Thanks!

∫εαçℏ · Accepted Answer

Well, you almost said them all already.

Suppose the denominator is AB⁴. A and B can't be simplified any more.
Then you split them such that you have A, B, B², B³, B⁴
For the numerators, you have a and b1, b2, b3, b4
The highest degree of a is one less than the highest of A; b one less than B
For both a and b, you need to run the degree down to 0, that is the constant term.

? · Answer

one million.) ?dx/(x^2 - one million) from 2 to 9 = one million/2*ln|(x - one million)/(x + one million)| eval. from 2 to 9 = one million/2 * ln(12/5) 2.) ?(x + 11)/(x^2 + 4x + 8) dx = ?(x + 2 + 9)/(x^2 + 4x + 8) dx = ?(x + 2)/(x^2 + 4x + 8) dx + ?9/((x + 2)^2 + 4) dx = one million/2*ln(x^2 + 4x + 8) + 9/4 * ?dx/(((x + 2)/2)^2 + one million) = one million/2*ln(x^2 + 4x + 8) + 9/2 * arctan(x/2 + one million) + C

Calc 2 How do I evaluate this Integral?

2 Answers