Need help with partial fraction.

In summary, the conversation discusses the process of finding the integral of a polynomial using partial fractions. The individual describes their attempt at solving the problem and asks for advice. The conversation also mentions a potential mistake and the need to solve for constants.
  • #1
zeppelinpage4
1
0

Homework Statement



I basically need to find the integral of (-5x^3-1x^2+4)/(x^4+2x^3)


Homework Equations



Just the basic rules associated with partial fractions when re-writing a polynomial with constants.

The Attempt at a Solution



Since the degree of the denominator is 4 and the degree of the numerator is 3 I went straight to factoring out the denominator.
I ended up getting
(x^4+2x^3)=x^3(x+2)

From this point I went forward with the basic steps (I think I made the mistake here, I'm not sure how to re-write the form for (x^3), so I broke it into (x^2) & (x)).

[(-5x^3-1x^2+4)/(x^4+2x^3)]= [(Ax+B)/(x^2)]+[C/x]+[D/(x+2)]

When I multiply both sides by (x^4+2x^3) (this is the denominator on the left side)

I end up with

(-5x^3-1x^2+4)=x^3(A+C+D) + x^2(2A+B+2C) + x(2B)

Normally i'd solve for the constants A, B, C and D to fit the polynomial on the left side BUT I don't know how to account for the "+ 4" at the end of (-5x^3-1x^2+4).

I know
A+C+D=-5
2A+B+2C=-1
2B=0

?=4

I'm very lost at how to approach this and any advice or help that could give me some clue would be very much appreciated. It's been a tough week with midterms and my calc teacher hasn't been helping. =P
 
Last edited:
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  • #2
This bit (x^4+2x^3)=x^3(x+2) is fine.

Personally I would look at the last two terms of the numerator, ask if they look anything familar. I find using that it falls out quite simply to something you can integrate. When you have got that, knowing the answer you will be able to check through your general method, which is what you have to use when you are not so lucky, and clear up where you went wrong maybe.

But there is virtue in simplicity, and I wouldn't present an unnecessarily laborious version.
 

1. What are partial fractions?

Partial fractions are a method of breaking down a complex rational expression into simpler fractions. This is done by expressing the denominator as a product of linear or irreducible quadratic factors and then finding the coefficients of the corresponding terms in the numerator.

2. When do we use partial fractions?

Partial fractions are often used in calculus and engineering when integrating rational functions. They can also be used in solving differential equations and in simplifying complex expressions.

3. How do you solve a partial fraction problem?

To solve a partial fraction problem, you first need to factor the denominator into linear or irreducible quadratic factors. Then, you set up a system of equations using the coefficients of the terms in the numerator and denominator. Finally, you solve for the unknown coefficients using algebraic methods.

4. What are the benefits of using partial fractions?

Using partial fractions can simplify complex rational expressions and make them easier to integrate or differentiate. It can also help in solving differential equations and in finding solutions to real-world problems in fields such as physics and engineering.

5. Are there any tips for solving partial fraction problems?

One tip for solving partial fraction problems is to start by factoring the denominator completely. Also, make sure to set up the system of equations correctly and carefully check your work to avoid any mistakes. Practice and familiarity with the method can also greatly improve your ability to solve these types of problems.

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