How to integ 2x^2 / (1-x^2) using partial fraction?

In summary, partial fraction decomposition is a method used to break down a complex fraction into simpler fractions with unique denominators. It is primarily used to simplify integration problems involving rational functions. To determine the partial fraction decomposition of a given fraction, you need to factor the denominator, set up a system of equations, and solve for unknown variables. An example of integrating a rational function using partial fraction decomposition is given by factoring the denominator, setting up an equation, solving for coefficients, and integrating each simpler fraction separately. However, there are restrictions to using partial fraction decomposition, such as the factorability of the denominator and the degree of the numerator.
  • #1
teng125
416
0
how to integ 2x^2 / (1-x^2)
using partial fracton??
 
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  • #2
ah...i forgot to type the answer=-2x + ln(1+x) - ln(1-x)
 
  • #3
First divide to get a "mixed" fraction: [itex]\frac{2x^}{1- x^2}= -2- \frac{2}{1-x^2}[/itex]. Now use partial fractions on the second term.
 
  • #4
Since the numerator and denominator are of the same order, you can turn it into a constant plus a fraction like so

[tex]\frac{2x^{2}}{1-x^{2}} = \frac{2x^{2}-2+2}{1-x^{2}} = \frac{-2(1-x^{2})}{1-x^{2}} +\frac{2}{1-x^{2}} = -2 +\frac{2}{1-x^{2}}[/tex]

Now use partial fractions on the second term to get your answer.
 

FAQ: How to integ 2x^2 / (1-x^2) using partial fraction?

What is partial fraction decomposition?

Partial fraction decomposition is a method used to break down a complex fraction into simpler fractions. This is done by expressing the complex fraction as a sum of simpler fractions with unique denominators.

Why is partial fraction decomposition used?

Partial fraction decomposition is used to simplify integration problems, especially when dealing with rational functions. It allows us to integrate each simpler fraction separately, making the overall integration process easier.

How do I determine the partial fraction decomposition of a given fraction?

To determine the partial fraction decomposition of a fraction, you need to follow a specific set of steps. First, factor the denominator of the fraction into irreducible factors. Then, set up a system of equations using the coefficients of each term in the numerator and solve for the unknown variables. Finally, substitute the values of the variables into the partial fraction decomposition form and combine like terms.

Can you provide an example of integrating a rational function using partial fraction decomposition?

Yes, for example, if we want to integrate 2x^2 / (1-x^2) using partial fraction decomposition, we would first factor the denominator to get (1-x)(1+x). Then, we would set up the equation 2x^2 / (1-x)(1+x) = A/(1-x) + B/(1+x) and solve for A and B. Once we have the values of A and B, we can substitute them into the partial fraction decomposition form and integrate each simpler fraction separately.

Are there any restrictions when using partial fraction decomposition?

Yes, there are a few restrictions to keep in mind when using partial fraction decomposition. First, the denominator of the original fraction must be factorable into linear and irreducible quadratic terms. Second, each irreducible quadratic factor must have a unique coefficient. Lastly, the degree of the numerator must be less than the degree of the denominator.

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