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Math Help - intergrals

  1. #1
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    Question intergrals

    integral of (dx)/(x(x^2+5))
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  2. #2
    Super Member angel.white's Avatar
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    Quote Originally Posted by Count Hesabu View Post
    integral of (dx)/(x(x^2+5))
    1. Use a partial fraction decomposition

    =\frac 15 \int\left( \frac 1x - \frac x{x^2+5} \right) ~dx

    2. Split up the integral

    =\frac 15 \int \frac 1x ~dx - \frac 15 \int \frac x{x^2+5} ~dx

    3. Make a smart substitution
    -----------------------------
    Substitute:
    a = x^2 +5

    \frac 12 ~da = x ~dx
    -----------------------------

    =\frac 15 \int \frac 1x ~dx - \frac 1{10} \int \frac 1{a} ~da

    4. Integrate

    =\frac 15 ln|x| - \frac 1{10} ln|a| +C

    5. Anti-substitute

    =\frac 15 ln|x| - \frac 1{10} ln|x^2+5| +C

    6. Square and square root the x on the left

    =\frac 15 ln|(x^2)^{1/2}| - \frac 1{10} ln|x^2+5| +C

    7. Pull the exponent into the coefficient

    =\frac 1{10} ln|x^2| - \frac 1{10} ln|x^2+5| +C

    8. Combine logs

    =\frac 1{10} ln\left|\frac {x^2}{x^2+5}\right| +C
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  3. #3
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    thanks it helped
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  4. #4
    Senior Member polymerase's Avatar
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    Quote Originally Posted by Count Hesabu View Post
    integral of (dx)/(x(x^2+5))
    \int \frac{1}{x(x^2+5)}\;dx=\int \frac{x^2-(x^2+5)}{x(x^2+5)}\;dx=\frac{1}{5}\int \frac{1}{x}\;dx-\frac{1}{5}\int \frac{x}{x^2+5}\;dx =\frac{1}{5}\ln|x|-\frac{1}{10}\ln|x^2+5|

    Thus, \int \frac{1}{x(x^2+5)}\;dx=\frac{1}{10}\ln\left(\frac{  x^2}{x^2+5}\right)+C

    Note: no need for absolute bars anymore because x^2 and x^2+5 will never be negative
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  5. #5
    Super Member angel.white's Avatar
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    Quote Originally Posted by polymerase View Post
    Note: no need for absolute bars anymore because x^2 and x^2+5 will never be negative
    Good point.
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