How to Solve an Antiderivative with Tricky Denominators

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Homework Help Overview

The problem involves finding the antiderivative of the function x^2 divided by the product of (x-1) and (x^2 + 4x + 5). The discussion centers around the complexities introduced by the denominators and potential methods for integration.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the initial attempts at manipulating the expression, including multiplying the denominator and considering the relationship between the numerator and the derivative of the denominator. There are suggestions of using partial fractions and completing the square as potential strategies.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts on possible methods such as partial fractions and substitutions. Some express uncertainty about the complexity of the problem, while others provide insights into potential approaches without reaching a consensus.

Contextual Notes

Participants question whether the problem requires showing work and discuss the challenges posed by the expression. There is mention of external resources that could assist in finding the antiderivative.

OSalcido
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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.
 
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OSalcido said:
[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. :frown:
 
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:

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

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.
 
Gib Z said:
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:

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

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.
 

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