# How to Solve an Antiderivative with Tricky Denominators

• OSalcido
In summary: It's always helpful to try out different methods to see which one works best. Keep at it, you'll get it eventually! In summary, the conversation involves trying to find the antiderivative of a complicated expression involving x^2 over a polynomial. The participants discuss various methods, including partial fractions and substitution, to solve the problem. Ultimately, it is suggested to try completing the square and using the arctan function to simplify the integral.
OSalcido

## 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.

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.

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.

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:

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

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.

## 1. What is an anti derivative?

An anti derivative, also known as the indefinite integral, is the inverse operation of differentiation in calculus. It is a mathematical function that, when differentiated, results in the original function.

## 2. How do you solve anti derivative problems?

To solve anti derivative problems, you must first identify the original function and then use integration techniques to find the anti derivative. These techniques include power rule, substitution, integration by parts, and trigonometric substitution.

## 3. What is the difference between a definite and indefinite integral?

A definite integral has specific limits of integration, while an indefinite integral does not. This means that a definite integral will result in a numerical value, while an indefinite integral will result in a function with a constant of integration.

## 4. Can you use anti derivatives to find area under a curve?

Yes, the definite integral, which is a type of anti derivative, can be used to find the area under a curve. This is known as the fundamental theorem of calculus.

## 5. How important is it to understand anti derivatives in science?

Understanding anti derivatives is crucial in many fields of science, particularly in physics and engineering. It is used to solve problems involving rates of change, such as velocity and acceleration, and to determine the total change over a given interval.

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