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Thread: Simple Integral question w/ parts

  1. #16
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    Quote Originally Posted by Chris L T521 View Post


    Where are you stuck in the division process?

    --Chris
    Well, I guess the whole thing, like how do you get $\displaystyle 1-\frac {1}{1+s^2}$
    It's something to do with the s^2 right?
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  2. #17
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by matt3D View Post
    Well, I guess the whole thing, like how do you get $\displaystyle 1-\frac {1}{1+s^2}$
    It's something to do with the s^2 right?
    You take $\displaystyle 1+s^2~|\overline{~~~~s^2}$

    We see that $\displaystyle s^2$ goes into $\displaystyle s^2$ 1 time.

    Thus, we should get

    $\displaystyle .~~~~~~~~~~~~~~~~~~{\color{blue}1}$
    $\displaystyle {\color{magenta}1+s^2}~|\overline{~~~~~~~~~s^2}$
    $\displaystyle .~~~~~~~~~\underline{-(1+s^2)}$
    $\displaystyle .~~~~~~~~~~~~~~~~~{\color{red}-1}$

    So our quotient will be $\displaystyle {\color{blue}1}+\frac{{\color{red}-1}}{{\color{magenta}1+s^2}}=1-\frac{1}{1+s^2}$, where $\displaystyle -\frac{1}{1+s^2}$ is the remainder.

    Does this make sense?

    --Chris
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  3. #18
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    This is done by polynomial long division. Check the link here to see how it is done: Polynomial Long Division
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  4. #19
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    If you want to do it quickly without polynomial division:

    $\displaystyle \frac{x^2}{x^2+1} $

    Add one/subtract one: $\displaystyle \frac{(x^2 + 1) - 1}{x^2+1}$
    Split: $\displaystyle \frac{x^2+1}{x^2+1} - \frac{1}{x^2+1}$
    Cancel: $\displaystyle 1 - \frac{1}{x^2+1} $
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  5. #20
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Chop Suey View Post
    If you want to do it quickly without polynomial division:

    $\displaystyle \frac{x^2}{x^2+1} $

    Add one/subtract one: $\displaystyle \frac{(x^2 + 1) - 1}{x^2+1}$
    Split: $\displaystyle \frac{x^2+1}{x^2+1} - \frac{1}{x^2+1}$
    Cancel: $\displaystyle 1 - \frac{1}{x^2+1} $
    Aha! I remember that trick now...I remember seeing it before...

    --Chris
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