# Anti derivative homework

## Homework Statement

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

## 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.

## Answers and Replies

JasonRox
Homework Helper
Gold Member
[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
Homework Helper
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
Homework Helper
Gold Member
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.