How to Solve an Antiderivative with Tricky Denominators

[OSalcido]

Homework Statement

Find the antiderivative of x^2 / [ (x-1)(x^2 + 4x +5)].

Homework Equations

The Attempt at a Solution

I first tried multiplying the denominator to get x^2 / x^3 + 3x^2 + x - 5 ... I noticed that the numerator is almost the derivative of the denominator, and if I can alter the expression to get du/u I can integrate it by using some form of ln |u|. I've tried several ways but I don't know where to go from here or what to do.

 

[JasonRox]

[I first tried multiplying the denominator to get x^2 / x^3 + 3x^2 + x - 5 ... I noticed that the numerator is almost the derivative of the denominator

Almost? I wouldn't say that. Not even close.

I thought about it, and it certainly doesn't look like an easy one to solve. I then put it in Maple to see what the solution looks like, and it's definitely looks tricky to get too.

What methods have you studied so far?

I think you can probably pull off partial fractions as your first steps actually. Oh, it looks doable now. I always think substitutions first. I'm still practicing my integrals.

 

[Gib Z]

You have some nasty stuff on your hands. Did it say to show working? If it just said "Find the antiderivative...", www.calc101.com can help :)

partial Fractions, i might be wrong, gave me this:

\frac{x^2}{(x-1)(x^2 +4x+5)} = \frac{9x+5}{10(x^2+4x+5)} + \frac{1}{10(x-1)}.

The 2nd bit is easy, the first bit isn't fun >.<.

Your going to have to do some completing the square, then let u = x+2 and hopefully our nice friend arctan will help you with the rest.

 

[JasonRox]

You have some nasty stuff on your hands. Did it say to show working? If it just said "Find the antiderivative...", www.calc101.com can help :)

partial Fractions, i might be wrong, gave me this:

\frac{x^2}{(x-1)(x^2 +4x+5)} = \frac{9x+5}{10(x^2+4x+5)} + \frac{1}{10(x-1)}.

The 2nd bit is easy, the first bit isn't fun >.<.

Your going to have to do some completing the square, then let u = x+2 and hopefully our nice friend arctan will help you with the rest.

Sounds like a solid plan to me.