# Rational function integral

1. Jan 8, 2005

### bomba923

How do I solve this integral?? (it's in the attachment)
(my-efforts.gif just explains a failed attempt)

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Last edited: Jan 8, 2005
2. Jan 8, 2005

### dextercioby

I've struggled with it and i couldn't come up with an answer.All i can tell u is that it cannot be factorized (decomposed in simple fractions) because it's not rational.The square in the denominator is the most annoying...

Daniel.

3. Jan 8, 2005

### bomba923

Wait..sorry neverm0nd----i found a solution!

Just use the Heaviside Method and make A, B, C--

substitute x= cos (theta)
and that dx= -sin (theta)
and then divide by the -sin (theta) as -(1-x^2)
-----------------------------
put A/((3-2x)^2)+B/((1+x)^(1/2))+C/((1-x)^(1/2),
then solve using heaviside

Last edited: Jan 8, 2005
4. Jan 8, 2005

### dextercioby

I told u it doesn't work...
$$\int \frac{d\theta}{(3-2\cos\theta)^{2}}$$
,via the substitution
$$\cos\theta\rightarrow x$$
becomes
$$-\int \frac{dx}{\sqrt{1-x^{2}}(3-2x)^{2}}$$
which is very irrational.

So your
$$\frac{A}{(3-2x)^{2}}+\frac{B}{\sqrt{1-x}}+\frac{C}{\sqrt{1+x}}$$

Daniel.

5. Jan 9, 2005

### Inquisitive_Mind

Use t method? (i.e. set t=tan(x/2), such that cos(x)=(1-t^2)/(1+t^2))

6. Jan 9, 2005

### marlon

This is the right way. I got to a solution like this. Besides the new differential shall be 2dt/(1+t²). Then you can manipulate this integral so that you will get two integrals. One of them is elementary (arcustangens)and the other one can be soved with partial integration...

Just try it...

remember with manipulation i mean something like this $$\frac {1+t^2}{(1+5t^2)^2} = \frac{1+ 5t^2}{(1+5t^2)^2} + \frac{-4t^2}{(1+5t^2)^2}$$

marlon

7. Jan 9, 2005

### marlon

And besides this decomposition is even wrong. You did not apply the rules correctly...But drop it because it is useless here and certainly with the sqrt you CANNOT do this !!!!!!!

marlon

8. Jan 9, 2005

### dextercioby

I knew the decomposition was wrong,i told him that substitution would lead nowhere.
Anyways,my result is
$$\int \frac{d\theta}{(3-2\cos\theta)^{2}}=\frac{4}{5}\frac{\tan\frac{\theta}{2}}{1+5\tan^{2}\frac{\theta}{2}}+\frac{6\sqrt{5}}{25}\arctan(\sqrt{5}\tan\frac{\theta}{2})+C$$

Daniel.

9. Jan 9, 2005

### marlon

You see, my trick worked...

marlon

10. Jan 9, 2005

### bomba923

Does the partial fractions technique work only for rational functions?
But yeah, my decomposition was wrong (now i see!)

11. Jan 9, 2005

### HallsofIvy

Staff Emeritus
Since only rational functions are fractions, yes, partial fractions only works for rational functions!