1. ## Integration Q

how does $\displaystyle \int\frac{3x}{3x-2}$ become
$\displaystyle \int \frac{2}{3x-2}+1$

2. ## Algebraic division

Hello vinson24
Originally Posted by vinson24
how does $\displaystyle \int\frac{3x}{3x-2}$ become
$\displaystyle \int \frac{2}{3x-2}+1$

$\displaystyle \frac{2}{3x-2}+1 = \frac{2}{3x-2}+ \frac{3x-2}{3x-2}$

$\displaystyle = \frac{2+(3x-2)}{3x-2}$

$\displaystyle =\frac{3x}{3x-2}$

The slightly more difficult trick is to work it the other way around. You do this by algebraic long division, starting with

$\displaystyle 3x-2\quad)\quad 3x = 1$, remainder something.

It's not very easy to set out the working using LaTeX, but it looks something like this:

$\displaystyle \begin{matrix} & &1& & \\ & &--&-&-\\3x-2&)&3x& & \\ & &3x&-&2\\ & &--&-&-\\ & & & &2\end{matrix}$

When you divide $\displaystyle 3x$ by $\displaystyle 3x-2$, then, the quotient is $\displaystyle 1$ and the remainder is $\displaystyle 2$. So the answer can be written:

$\displaystyle \frac{3x}{3x-2} = 1 +\frac{2}{3x-2}$

3. i see it now thanks alot

4. Originally Posted by vinson24
how does $\displaystyle \int\frac{3x}{3x-2}$ become
$\displaystyle \int \frac{2}{3x-2}+1$
Divide 3x by 3x- 2. What is the quotient? What is the remainder?

5. Originally Posted by vinson24
how does $\displaystyle \int\frac{3x}{3x-2}$ become
$\displaystyle \int \frac{2}{3x-2}+1$
if polynomial long division isn't your thing, note that some algebraic manipulation can get you there as well

$\displaystyle \frac {3x}{3x - 2} = \frac {3x - 2 + 2}{3x - 2} = \frac {3x - 2}{3x - 2} + \frac 2{3x -2} = 1 + \frac 2{3x - 2}$