# Homework Help: Why does (3x-5)/(x^2+5x+8) = (3(2x+5))/(2(x^2+5x+8))-25/(2(x^2+5x+8))?

1. Aug 7, 2011

### GreenPrint

I don't understand why
(3x-5)/(x^2+5x+8) = (3(2x+5))/(2(x^2+5x+8))-25/(2(x^2+5x+8))

and was hopping somebody could explain why it's not partial fractions or anything is it?

2. Aug 7, 2011

### HallsofIvy

Well, except for the "2"s on the right, the denominators are exactly the same aren't they?
So this is A/D= B/(2D)- C/(2D)= (1/2)(B/D- C/D)= (1/2)((B-C)/D).
Is "A" in this case the same as (1/2)(B- C)?

That is, what is (3/2)(2x+ 5)- (25/2)?

3. Aug 7, 2011

### GreenPrint

Huh... Yes the denominators are the same.

A/D= B/(2D)- C/(2D)= (1/2)(B/D- C/D)= (1/2)((B-C)/D)

This is essentially splitting the rational function into two terms that are half the original over the same denominator and finding out the terms in the numerator for each half?

Yes A = (1/2)(B- C)

(3/2)(2x+ 5)- (25/2) = 3x - 5

huh interesting I never though of doing this...

The reason why I asked this was because I was trying to evaluate

integral (x^2+2)/(x(x^2+5x+8)) dx

and got to this point

1/4 ln|x| + c + 1/4 integral (3x-5)/(x^2+5x+8) dx

do you know of any other way of evaluating

integral (3x-5)/(x^2+5x+8) dx besides splitting (3x-5)/(x^2+5x+8) into (3(2x+5))/(2(x^2+5x+8))-25/(2(x^2+5x+8)) and taking two separate integrals because I don't think I would of come up with that on my own in like the middle of a test

4. Aug 7, 2011

### GreenPrint

I tried all sorts of things

x^2 + 5x + 8 can be expressed by completing the square as (x+5/2)^2+7/4

and you can than proceed to use a trig sub but every time i do it gets overly complicated

5. Aug 7, 2011

### ArcanaNoir

where did you find this integral?

6. Aug 7, 2011

### GreenPrint

My textbook in
Chapter 7 integration Techniques - Section 4 Partial Fractions

7. Aug 7, 2011

### ArcanaNoir

but it doesn't do the partial fraction thing
:(

Name of textbook and problem number please.

8. Aug 7, 2011

### GreenPrint

ya i know i got up to here

(x^2+2)/(x(x^2+5x+8)) = 1/(4x) + (3x-5)/(4(x^2+5x+8))

integral of the first term is just 1/4 ln(x) + c

1/4 ln(x) + c + 1/4 integral (3x-5)/(x^2+5x+8)

I then completed the square and tried to do a trig sub

9. Aug 7, 2011

### GreenPrint

This is what I got when I tried to do a trig sub

-3 ln|sqrt(7)/(2 sqrt((x+5/2)^2+7/4))| + (10*sqrt(7))/7*cot^(-1)((2x+3)/sqrt(7))

10. Aug 7, 2011

### GreenPrint

1/4 ln(|x|)-3/4 ln(|sqrt(7)/(2 sqrt((x+5/2)^2+7/4))|) + (10*sqrt(7))/28*cot^(-1)((2x+3)/sqrt(7))+c

Last edited: Aug 7, 2011
11. Aug 7, 2011

### GreenPrint

that's what i get and im trying to see if it's correct

12. Aug 7, 2011

### GreenPrint

I'll post this in calculus and beyond sense this is now mostly calculus help

13. Aug 7, 2011

### ArcanaNoir

OMG Sorry I got distracted by PHD! I'm back on the case now! (In the calc sections.)

14. Aug 7, 2011

### ArcanaNoir

No, you know what? This integral BITES. Save it for your professor. Make him squirm. :P

15. Aug 7, 2011

### SammyS

Staff Emeritus
Maybe you could come up with this.

The derivative of the denom. is 2x+5.

Just multiply the numerator by 2/3 and add ( and subtract) the needed constant term -- to the numerator also.

$\displaystyle \frac{3}{2}\frac{(2/3)(3x-5)}{x^2+5x+8}=\frac{3}{2}\frac{2x-\frac{10}{3}+5 -5}{x^2+5x+8}$
$\displaystyle =\frac{3}{2}\frac{2x+5}{x^2+5x+8}+\frac{3}{2}\frac{-\frac{10}{3}-5}{x^2+5x+8}$​

The first term gives a log when integrated.

Clean up the second term, of course the numer. is -25, then complete the square in the denom. & integrate to get an arctan.

16. Aug 7, 2011

### GreenPrint

I solved it see my thread in calculus and beyond section i decided to post it there because now that i understood the basic math i needed help in calculus... it did suck but i was able to solve it lol

17. Aug 7, 2011

hmm thanks