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Finding Integral of a Divided Function (Hyperbola?)

  1. Sep 21, 2012 #1
    1. The problem statement, all variables and given/known data
    Integrate between y=0 and y=-0.5:
    ∫((y+0.5)/(0.5-y)) dy

    2. Relevant equations
    Can you please show me how to integrate it...Then I will be able to take it from there and substitute in the appropriate values.


    3. The attempt at a solution
    Quotient rule in reverse?...wouldn't work
    Using ln ...wouldn't work
    Try to factorise...nope
    Use rationalising of denominator (conjugate idea)...that just makes it even more complicated
    Now I have run out of ideas ...this is out of my scope. (Scholarship paper NZ)

    Thank you in advance,
    Kindest Regards.
     
  2. jcsd
  3. Sep 21, 2012 #2

    LCKurtz

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    Hint: Divide the numerator by the denominator to make the degree of the numerator one less than the degree of the denominator.
     
  4. Sep 21, 2012 #3
    Hmmm...Thanks for the hint, but I don't get what you meant here...
    Isn't the numerator already being divided by the denominator and aren't they to the same degree anyway?
     
  5. Sep 21, 2012 #4

    LCKurtz

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    Write it as$$
    -\frac{y + \frac 1 2}{y -\frac 1 2}$$
    and do one step of a long division writing it as quotient + remainder/divisor.

    Alternatively you could write it as$$
    -\frac {(y-\frac 1 2)+1}{(y-\frac 1 2)}$$and break it into two terms. It's the same result either way.
     
  6. Sep 21, 2012 #5

    SammyS

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    Yes, they're the same degree, so use some sort of division algorithm to find a quotient & remainder.
     
  7. Sep 21, 2012 #6
    Nice! I like the alternative (because its the only thing I understood) from LCKurtz...rationalising the denominator.
    The rest of it didn't make much sense...how can you integrate with a remainder! (I must be missing something)
    Thank you so much for all your help.

    P.S.: Sorry for not being able to understand what you guys said.

    Kindest Regards
     
  8. Sep 21, 2012 #7

    SammyS

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    Here's how that remainder works:

    Write the remainder of 1 as a fraction.

    [itex]\displaystyle \frac{1}{y-\frac{1}{2}}[/itex]

    So that [itex]\displaystyle -\frac {(y+\frac 1 2)}{(y-\frac 1 2)}=-\left(1+\frac{1}{y-\frac{1}{2}}\right)[/itex]

    Although expressions like the one here, I often use the same method as LCKurtz showed you. For more complicated situations, you should know how to express the remainder as a fraction.
     
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