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Math Help - Integration by Partial Fractions

  1. #1
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    Integration by Partial Fractions

    From 0 to 1, integrate x/(x^2+4x+13). I never could understand partial fractions but I have got a few but this one is throwing me off.
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  2. #2
    Member Nacho's Avatar
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    But In this example I dont use parcial fraction, in fact this integral is an arctan and a logarythm
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  3. #3
    Senior Member polymerase's Avatar
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    \int\frac{x}{x^2+4x+13}=\int\frac{x+2}{x^2+4x+13}\  ;dx-2\int\frac{dx}{x^2+4x+13}

    complete the square on the second term and make the appropriate substitutions you should end up with this:

    \frac{1}{2}\int\frac{du}{u}-2\int\frac{dx}{(x+2)^2+9}

    second term, you should note, is a standard arctan

    Thus finally, \int\frac{x}{x^2+4x+13}\;dx=\frac{1}{2}\ln(x^2+4x+  13)-\frac{2}{3}\arctan\left({\frac{x+2}{3}}\right)+C
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