Q. Solve ∫ (5x^2 - 42x + 24)/(x^3 - 10x^2 + 12x + 72) dx
∫ x^n dx = ∫ (1/n+1)x^(n+1) + C
∫ 1/x dx = ln x + C
The Attempt at a Solution
Here's what I've done so far:
(i) I factored the bottom equation into (x + 2)(x - 6)^2
and then separated the fraction 1/[(x + 2)(x - 6)^2] into a sum of three fractions of the form:
A/(x + 2) + B/(x - 6) + C/[ (x-6)^2 ]
and found that A = 1/64, B = -1/64, C = 1/8
(ii) The only way I can see to go from here is to split the top integral into three integrals:
(1/64) ∫ (5x^2 - 42x + 24)/(x + 2) dx + (-1/64) ∫ (5x^2 - 42x + 24)/(x - 6) dx + (1/8) ∫ (5x^2 - 42x + 24)/(x-6)^2 dx
and solve them all individually using substitution, letting u = x+2, x-6 and x-6 respectively.
The answer I get in this way is: (5/8)x - 21/4 +2ln(x+2) + 3ln(x-6) + 6/(x-6) + C, C = constant of integration,
which is incorrect, according to the WebWork site, but I can't find any algebraic errors.
So, I'm thinking perhaps what I do in step (ii) is not the best way to do it.
Has anyone any suggestions?
Thanks for any help!