Indefinite Integral of (3x^2-10)/(x^2-4x+4) dx using PARTIAL FRACTION

In summary, the student tried using A/(x-2) + B/(x-2)^2 and C/(x-2)^2 but they didnt get a coeffecient of an x^2. They also tried using the usual A/(x-2) + B/(x-2)^2 but that didnt work either. They found out that if the degree of the numerator and denominator are the same, then long division is the only option.
  • #1
Susie babe
3
0

Homework Statement



Integrate (3x^2-10)/(x^2-4x+4) dx using partial fractions.


Homework Equations



None

The Attempt at a Solution



I tried using A/(x-2) + B/(x-2)^2 but I didnt get a coeffecient of an x^2.

I've also tried using (Ax+B)/(x-2) + C/(x-2)^2



Though I honestly thought that the usual A/(x-2) + B/(x-2)^2 would work, How do I know which formula to use. I know you have to do long division if the numerator has an x to a larger value than that of the denominator. I know when you have something like: (x^2-1) or (x^3+2) in the denominator then you use the Ax+B, Cx+D formula.
 
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  • #2


Susie babe said:

The Attempt at a Solution



I tried using A/(x-2) + B/(x-2)^2 but I didnt get a coeffecient of an x^2.

I've also tried using (Ax+B)/(x-2) + C/(x-2)^2
Though I honestly thought that the usual A/(x-2) + B/(x-2)^2 would work, How do I know which formula to use.
The last thing that you said is correct...
[tex]\frac{A}{x - 2} + \frac{B}{(x - 2)^2}[/tex]
... because the denominator is a linear factor squared. But before you try partial fractions, you have to use long division because the degrees of the numerator and denominator are the same.
 
  • #3


The degree of the numerator is equal to the degree of the denominator. Try long division before partial fraction decomposition.
 
  • #4


Susie babe said:

Homework Statement



Integrate (3x^2-10)/(x^2-4x+4) dx using partial fractions.


Homework Equations



None

The Attempt at a Solution



I tried using A/(x-2) + B/(x-2)^2 but I didnt get a coeffecient of an x^2.

I've also tried using (Ax+B)/(x-2) + C/(x-2)^2



Though I honestly thought that the usual A/(x-2) + B/(x-2)^2 would work, How do I know which formula to use. I know you have to do long division if the numerator has an x to a larger value than that of the denominator. I know when you have something like: (x^2-1) or (x^3+2) in the denominator then you use the Ax+B, Cx+D formula.

Re-write the numerator as
[tex] 3x^2-10 = 3(x^2-4x+4)+12x - 22.[/tex]
 
  • #5
Ah, so if the degree of the numerator and that of the denominator are the same then you have to use long division, didnt know that. Thanks a lot guys it worked out well.
 

1. What is partial fraction decomposition?

Partial fraction decomposition is a method used to break down a rational function into simpler fractions. This allows us to integrate the function more easily.

2. Can any rational function be decomposed into partial fractions?

Yes, any proper rational function (where the degree of the numerator is less than the degree of the denominator) can be decomposed into partial fractions.

3. How do you decompose a rational function into partial fractions?

To decompose a rational function, we first factor the denominator into linear and irreducible quadratic factors. Then, we set up an equation with unknown coefficients for each factor and solve for them using algebraic manipulation.

4. How does partial fraction decomposition help with integrating a function?

After decomposing a rational function into partial fractions, we can integrate each simpler fraction separately. This allows us to break down a complicated integration problem into smaller, more manageable ones.

5. Are there any special cases to consider when using partial fraction decomposition?

Yes, there are a few special cases to consider, such as repeated factors in the denominator and complex roots. These cases require additional steps in the decomposition process.

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