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  1. #1
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    Integration Q

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

    Hello vinson24
    Quote Originally Posted by vinson24 View Post
    how does $\displaystyle \int\frac{3x}{3x-2}$ become
    $\displaystyle \int \frac{2}{3x-2}+1$
    The short answer is that

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

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

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

    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:

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

    Grandad

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  3. #3
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    i see it now thanks alot
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    Quote Originally Posted by vinson24 View Post
    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?
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by vinson24 View Post
    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

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