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Math Help - Integration Problem

  1. #1
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    Integration Problem

    \int_{0}^{\sqrt3} \frac{5x^2 dx}{x^2 +1}

    Not entirely sure how to do this one.
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  2. #2
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    Quote Originally Posted by looseenz2 View Post
    \int_{0}^{\sqrt3} \frac{5x^2 dx}{x^2 +1}

    Not entirely sure how to do this one.
    Note that \frac{5x^2}{x^2 + 1} = 5 \left( \frac{x^2}{x^2 + 1}\right) = 5 \left( \frac{(x^2 + 1) - 1}{x^2 + 1}\right) = 5 - \frac{5}{x^2 + 1}.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Note that \frac{5x^2}{x^2 + 1} = 5 \left( \frac{x^2}{x^2 + 1}\right) = 5 \left( \frac{(x^2 + 1) - 1}{x^2 + 1}\right) = 5 - \frac{5}{x^2 + 1}.
    Could you please explain what you did between the last two steps? I'm not entirely sure what you did.
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  4. #4
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    Quote Originally Posted by looseenz2 View Post
    Could you please explain what you did between the last two steps? I'm not entirely sure what you did.
    the basic rule is ...

    when the numerator has a degree \ge the degree of the denominator, divide the fraction.

    this can be done with long division (as shown below), or the "shortcut" shown by mr. F

    Code:
    ........1
          ----------
    x^2+1 | x^2 
    ........x^2 + 1
    .....  -----------
    ........... - 1
    the quotient is 1 + \frac{-1}{x^2+1} = 1 - \frac{1}{x^2+1}<br />
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