Integration by partial fractions.

  • #1
1.[tex]\int[/tex]2x^2+x+9/(9x+1)(x^2+9) dx

2. (A/9x+1) + [(Bx + C ) / (x^2 + 9)]

I get the worst numbers when I solve the system. The question is from an old exam and calculators are not allowed. Am I doing something wrong or is there another way to integrate this?
 

Answers and Replies

  • #2
35,118
6,857
Are you sure of the numbers you are getting? IOW, did you check to make sure that your two separate fractions add up to the larger one?
 
  • #3
Yes, I used a solver to check my work because it did not seem right. I am fairly sure the fractions are broken up properly as well. =(
 
  • #4
35,118
6,857
Yes, your decomposition is fine. Out of curiosity, what did you get for A, B, and C?
 
  • #5
a=37/41 b=5/41 c = 7/41 I mean its not that outrageous I guess. I guess I just assume I do things wrong lol.
 
  • #6
vela
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Just to make sure, the integrand is

[tex]\frac{2x^2+x+9}{(9x+1)(x^2+9)}[/tex]

and not

[tex]2x^2+x+\frac{9}{(9x+1)(x^2+9)}[/tex]

right? In either case, your values for A, B, and C aren't correct. If the first version of the integrand is the right one, the coefficients come out even less pretty.
 
  • #7
Yes the first version is correct. -.- Do you have any suggestions?
 
  • #8
2,981
5
your partial fractions expansion is:

[tex]
\frac{A}{9 x + 1} + \frac{B \, x + C}{x^{2} + 9}
[/tex]

Multiplying both sides by [itex](9 x + 1)(x^{2} + 9)[/itex] and comparing similar terms, you will get 3 equations for 3 unknowns.
 
  • #9
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I get the following system:

[tex]
\left\{
\begin{array}{rcl}
9 A + C & = & 9 \\

B + 9 C & = & 1 \\

A + 9 B & = & 2
\end{array} \right.
[/tex]
 
  • #10
vela
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After you multiply by the denominator, I'd set x=-1/9, which allows you to solve for A, then use the equations Dickfore got to solve for B and C.
 
  • #11
Yeah.... It must be some type of cruel joke.
 
  • #12
2,981
5
The denominators for the solutions are equal to the number of days in a year.
 
  • #13
361/365 41/365 36/365
It was on an old exam lol. I hope there's nothing like that on our exam =O.
 

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