Integration of rational functions by partial fractions

In summary, the conversation is about finding the partial fraction decomposition of the function x^2/(x^2 + x + 2). The person performing the task mentions that they performed polynomial division and obtained the expression 1 + (-x-2)/(x^2 + x + 2). They also mention that the denominator may not be factorable due to a negative discriminant. The book may be asking a trick question and the simplified form of the decomposition might just be (Ax + B)/(x^2 + x + 2).
  • #1
vande060
186
0

Homework Statement



write out the form of the partial fraction decomposition of the function, do not determine the numerical values of the coefficients

x^2/(x^2 + x + 2)



Homework Equations





The Attempt at a Solution



since the numerator is not less of a degree than the denominator I preformed polynomial division to obtain:

1 + (-x-2)/(x^2 + x + 2)

here I don't think the denominator is even factorable, because the discriminant (b^2 - 4ac) is < 0. . Is the book asking me a trick question here?

EDIT: is it just simply (Ax + B)/(x^2 + x +2)
 
Last edited:
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  • #2
Yes, if you are referring to (-x-2)/(x2 + x + 2) .
 
  • #3
SammyS said:
Yes, if you are referring to (-x-2)/(x2 + x + 2) .


thanks!
 

FAQ: Integration of rational functions by partial fractions

What is integration of rational functions by partial fractions?

Integration of rational functions by partial fractions is a method used to simplify and solve integrals of rational functions, which are fractions made up of polynomials in the numerator and denominator. It involves breaking down the rational function into simpler fractions, called partial fractions, and then integrating each part separately.

Why is integration of rational functions by partial fractions useful?

This method is useful because it allows us to solve integrals that would otherwise be difficult or impossible to solve using other methods. It also helps us to find the integral of a rational function in a more efficient manner.

What is the process of integrating rational functions by partial fractions?

The process involves breaking down the rational function into partial fractions, which are simpler fractions with linear denominators. We then find the constants that make up each partial fraction and integrate them separately. Finally, we combine the individual integrals to find the overall integral of the original rational function.

When should integration of rational functions by partial fractions be used?

This method should be used when the degree of the numerator is less than the degree of the denominator, or when the denominator can be factored into linear factors. It is also useful when the rational function has repeated linear factors or non-real roots.

What are some tips for solving integrals using integration of rational functions by partial fractions?

Some helpful tips include factoring the denominator completely, using the method of undetermined coefficients to find the constants, and checking for any potential algebraic errors. It is also important to remember to always check your answer by taking the derivative to make sure it matches the original function.

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