Integrating Partial Fractions: x^2+2x-1/2x^3+3x^2-2x (x>1/2)

In summary, The conversation discusses integration by partial fraction and solving the antiderivative of a given function with the restriction that x is greater than 1/2. The reason for this restriction is due to the function not being continuous for values less than 1/2. The conversation also mentions solving for A, B, and C in order to integrate. The conversation ends with gratitude for the explanation.
  • #1
oyala
3
0
Hi guys I have a question here relating integration by partial praction..


the question said what is the antiderivative of

x^2+2x-1/2x^3 +3x^2 - 2x

valid only when x > 1/2.

anyway i had poor background in math and working hard to catch up...
I don't understant why "valid only when x > 1/2".
if you integrate that indefinte integral when do you have to do to make sure relate x>1/2
 
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  • #2
While the function certainly exists for many points less than 1/2, it is not continuous over the interval -inf<x<1/2 since it has three singularities (I'll let you solve for these). Thus it is best to define the domain of the function as 1/2<x<inf. Now as for the integration by partial fraction, try solving this equation for A, B and C by setting x to certain values(in order to eliminate only A, B or C):

[tex]\frac{x^{2}+2x-1}{2x^{3}+3x^{2}-2x}=\frac{A}{x}+\frac{B}{x+2}+\frac{C}{2x-1}[/tex]

Then you'll be able to integrate.
 
  • Like
Likes Nour elden
  • #3
hey thank you so much.
you explain it better than what I thought


great work !
 
  • #4
Anytime man, I appreciate the feedback.
 

Related to Integrating Partial Fractions: x^2+2x-1/2x^3+3x^2-2x (x>1/2)

1. What is partial fraction decomposition?

Partial fraction decomposition is a method used to break down a fraction into smaller fractions with simpler denominators. It is commonly used in integration to simplify complex fractions before integrating.

2. How do you solve this specific partial fraction problem?

To solve this problem, you first need to factor the denominator. In this case, the denominator is 2x^3+3x^2-2x, which can be factored into 2x(x+1)(x-1). Once factored, you can set up the equation A/2x + B/(x+1) + C/(x-1) = (x^2+2x-1)/(2x(x+1)(x-1)) and solve for the unknown constants A, B, and C using algebraic techniques.

3. Can you use partial fraction decomposition for any fraction?

No, partial fraction decomposition can only be used for proper rational functions, which are fractions where the degree of the numerator is less than the degree of the denominator. Fractions with equal degrees in the numerator and denominator require different methods for integration.

4. How do you know when to use partial fraction decomposition in integration?

Partial fraction decomposition is typically used when integrating a rational function with a denominator that cannot be easily integrated using other methods, such as u-substitution or trigonometric substitution. It can also be used when the degree of the numerator is equal to or greater than the degree of the denominator.

5. Are there any shortcuts or tricks for solving partial fraction problems?

There are a few techniques that can make solving partial fraction problems easier, such as using the method of undetermined coefficients or grouping terms with similar denominators. However, ultimately it requires knowledge of algebraic manipulation and practice to effectively solve partial fraction problems.

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