Finding Constants for Partial Fraction Integration

In summary, a partial fraction integral is a mathematical technique used to simplify and solve integrals involving rational functions. It is necessary when the integrand is a rational function, and it involves breaking down the function into simpler fractions using partial fraction decomposition. This process requires factoring the denominator and solving for constants. Partial fraction integrals can be used for improper integrals, but they have limitations as they can only be applied to rational functions and can become complex for certain types of functions.
  • #1
gordda
20
0
hello, i have the Intergal ∫(5x^2-3x+1)/((x^2+1)(x-2)) dx
i do not know what to do first, i firstly tried to change to partial fraction but that didn't work to, then i tried substitution technique and again i didn't have anything to substitue that got me any further, i don't know what else to try, somebody please help.
thanx:)
 
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  • #2
Show us your partial fractions work.
 
  • #3
Determine constants A,B,C so that:
[tex]\frac{5x^{2}-3x+1}{(x^{2}+1)(x-2)}=\frac{Ax+B}{x^{2}+1}+\frac{C}{x-2}[/tex]
 

FAQ: Finding Constants for Partial Fraction Integration

What is a partial fraction integral?

A partial fraction integral is a mathematical technique used to simplify and solve integrals involving rational functions. It involves breaking down a complex fraction into simpler fractions, which can then be integrated using standard integration techniques.

When is the use of partial fraction integrals necessary?

Partial fraction integrals are necessary when the integrand (the expression being integrated) is a rational function, meaning it is a fraction with polynomials in the numerator and denominator. In order to integrate the rational function, it must be broken down into simpler fractions using partial fraction decomposition.

How do you decompose a rational function into partial fractions?

To decompose a rational function into partial fractions, the denominator of the fraction must first be factored into its irreducible factors. Then, each factor will correspond to a term in the decomposition, with a unique constant as the numerator. These constants can be solved for by setting the original fraction equal to the partial fraction decomposition and solving for each constant.

Can partial fraction integrals be used for improper integrals?

Yes, partial fraction integrals can be used to solve improper integrals. In this case, the improper integral must be rewritten as a rational function, and then the partial fraction decomposition can be used to integrate it.

Are there any limitations to using partial fraction integrals?

Partial fraction integrals can only be used for rational functions, so they are not applicable for other types of integrals. Additionally, the decomposition and integration process can become complex and difficult for certain types of rational functions, making it a less efficient method in those cases.

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