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Math Help - Intergrate Calc dont you love it

  1. #1
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    Intergrate Calc dont you love it

    Intergrate (2x^2+4x+22) / (x^2+2x+10) dx
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  2. #2
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    Hello, Bust20001

    \int\frac{2x^2+4x+22}{x^2+2x+10}\,dx

    Divide: . \int\left(2 + \frac{2}{x^2 +2x+10}\right)\,dx

    . . . . . =\;\;\int\left(2 + \frac{2}{(x+1)^2+9}\right)\,dx

    . . . . . = \;\;2x + \frac{2}{3}\arctan\left(\frac{x+1}{3}\right)+ C

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  3. #3
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    Quote Originally Posted by Bust2000 View Post
    Intergrate (2x^2+4x+22) / (x^2+2x+10) dx
    \frac{2x^2 + 4x + 22}{x^2 + 2x + 10} = 2\left( \frac{x^2 + 2x + 11}{x^2 + 2x + 10} \right) = 2\left( \frac{(x^2 + 2x + 10) + 1}{x^2 + 2x + 10} \right)


    = 2\left( 1 + \frac{1}{x^2 + 2x + 10} \right) = 2\left( 1 + \frac{1}{(x + 1)^2 + 9} \right) = \left( 1 + \frac{1}{3} \, \frac{3}{(x + 1)^2 + 3^2} \right).

    You should have better luck with this form .....
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  4. #4
    Junior Member roy_zhang's Avatar
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    Quote Originally Posted by Bust2000 View Post
    Intergrate (2x^2+4x+22) / (x^2+2x+10) dx
    When your integrand is in the form \frac{P(x)}{Q(x)}, where P(x),Q(x) are both polynomials, and the order of P(x) is greater or equal to the order of Q(x); always try long division first before integrating.

    Here we have:
    \frac{2x^2+4x+22}{x^2+2x+10} = 2+\frac{2}{x^2+2x+10}

    Now,
    \int\frac{2x^2+4x+22}{x^2+2x+10}dx = \int (2+\frac{2}{x^2+2x+10})dx = \int2dx + \int[\frac{2}{(x+1)^2+3^2}]dx

    use u substitution (let u= x+1)

    we got:

    \int \frac{2x^2+4x+22}{x^2+2x+10}dx=2x + \frac{2}{3}\arctan(\frac{x+1}{3}) + C
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