# Thread: integration of quotient

1. ## integration of quotient

if anyone could help me with the integration of this quotient it would be greatly appreciated.

i have tried and tried but just cant get the right integral

$
\int \frac{x+2}{x^2-7x+12}dx
$

thanks a lot

2. Originally Posted by rudy
if anyone could help me with the integration of this quotient it would be greatly appreciated.

i have tried and tried but just cant get the right integral

$\int \frac{x+2}{x^2 -7x +12} \, dx$

thanks a lot

The technique to use is partial fraction decomposition.

If you show you're working it will be easier to give you the appropriate help.

3. yea sorry bout the triple posting earlier.

umm ok we have only been taught the substitution method.
where you use "u" etc

4. Oooooppsss. Completely misread the fraction. Sigh.

5. Originally Posted by rudy
yea sorry bout the triple posting earlier.

umm ok we have only been taught the substitution method.
where you use "u" etc
Well, in that case:

Multiply by 2 and divide by 2: $\frac{1}{2} \int \frac{2x+4}{x^2-7x+12}dx$

Add 11 and subtract 11: $\frac{1}{2} \int \frac{2x+4+11-11}{x^2-7x+12}dx$

The reason why I did this is to get one integral to be of the form:
$\int \frac{f'(x)}{f(x)} dx$

Simplify and split the integrals: $\frac{1}{2} \int \frac{2x-7}{x^2-7x+12}dx + \frac{11}{2} \int \frac{1}{x^2-7x+12}dx$

The first integral should be easy. In the second one, complete the square, then try to get it in the form of:
$\frac{1}{u^2+1}$

which should be a familiar standard form to you.

6. All the obvious (and frightfully easy!) methods have already been done and/or mentioned so I decided to post this long winded one:

Use a substitution for the last bit (u=x-3) or you can try some more "adding 0" on the numerator.

7. ^ Very cool method.