Help finishing ∫ (x - 1)/(x^2 - 4x + 5)?
∫ (x - 1)/(x^2 - 4x + 5)
I know that u = x - 2
From there I think this is ∫ (u + 1)/(u^2 + 1)
Help???
Update:Thank you very much. I am lost in my own notes on the computer. This has helped alot. I think I will try working more of these types on actual paper instead of the computer.
More Questions From This User See All
Answers & Comments
Verified answer
How do you know that u = x-2 ?
It works as a valid form of substitution... but how did you arrive at it?
So, you are at this expression now:
∫ (u + 1)/(u^2 + 1) du
Split it up into fractions
∫ u / (u^2 + 1) + 1 / (u^2 + 1) du
∫ u / (u^2 + 1) du + ∫ 1 / (u^2 + 1) du
The left integral is
∫ u / (u^2 + 1) du = ½ ln(u^2 + 1)
The right integral is
∫ 1 / (u^2 + 1) du = arctan(u)
So your integral is
∫ u / (u^2 + 1) + 1 / (u^2 + 1) du = ½ ln(u^2 + 1) + arctan(u) + C
Plug your u = x-2 back in
½ ln((x-2)^2 + 1) + arctan(x-2) + C
½ ln(x^2 - 4x + 5) + arctan(x-2) + C