Integration by Partial Fractions

In summary, the conversation discusses how to solve the integral ∫ 4x/(x^3+x^2+x+1) dx. The participants mention that the degree of the numerator is less than the denominator, making long division impossible. They also suggest looking at the denominator as a geometric sum and using partial fractions. Finally, one participant mentions trying to factor the denominator using the rational root theorem, which ultimately leads to solving the problem.
  • #1
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Homework Statement



∫ 4x/(x^3+x^2+x+1) dx

The Attempt at a Solution



I really don't know where to start, you can't complete the square, the degree of the numerator is less than the denominator so you can't use long division to simplify it.

I can't really simplify the denominator as well, so I am stuck.

Help is greatly appreciated.!

Edit: I don't need you to work out the whole problem, i just need help getting started.
 
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  • #2
If you will look hard at the denominator you will see that it is a geometric sum. You can replace that long expression with something simpler, which is the sum of the terms. Not sure what that would be? Look up geometric sum.

At that point maybe partial fractions will help.
 
  • #3
KTiaam said:

Homework Statement



∫ 4x/(x^3+x^2+x+1) dx


The Attempt at a Solution



I really don't know where to start, you can't complete the square, the degree of the numerator is less than the denominator so you can't use long division to simplify it.

I can't really simplify the denominator as well, so I am stuck.

Help is greatly appreciated.!

Edit: I don't need you to work out the whole problem, i just need help getting started.

Try factoring the denominator. The rational root theorem says that there are only two possibilities for rational roots - and one of them does work.
 
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  • #4
Solved. Thank you Mark
 

1. What is integration by partial fractions?

Integration by partial fractions is a method used to integrate rational functions, which are fractions with polynomials in the numerator and denominator. It involves breaking down a complex fraction into simpler fractions, making it easier to integrate.

2. When is integration by partial fractions used?

This method is used when trying to integrate a rational function, especially when the degree of the numerator is less than the degree of the denominator. It is also used when the denominator of the fraction can be factored into linear and irreducible quadratic factors.

3. What are the steps for integration by partial fractions?

The first step is to factor the denominator of the rational function. Then, using the factored form, write the rational function as a sum of simpler fractions. Next, determine the unknown coefficients by equating the numerators of the simpler fractions to the original numerator. Finally, integrate each of the simpler fractions and combine the results to get the final solution.

4. Are there any restrictions when using integration by partial fractions?

Yes, there are a few restrictions to keep in mind. The denominator of the original rational function cannot have repeated linear factors, and all the irreducible quadratic factors must be distinct. Also, if the degree of the numerator is equal to or greater than the degree of the denominator, then the fraction cannot be broken down using this method.

5. How can integration by partial fractions be applied in real-life situations?

Integration by partial fractions can be used in various engineering and physics applications, such as circuit analysis, control systems, and fluid mechanics. It can also be used in economics and finance to model and analyze interest rates and investment growth.

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