What is the integral of (x^2)/(x^4-20x^2-125) ?

The integral of (x^2)/(x^4-20x^2-125) is 1/(6sqrt(5))arctan(x/(sqrt(5)))-1/12 ln|x+5|+1/12 ln|x-5|+C

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integral of (x^2)/(x^4-20x^2-125)

\int \frac{x ^{2}}{x ^{4} - 20 x ^{2} - 125} d x

\frac{1}{6 5} arctan (\frac{x}{5}) - \frac{1}{12} ln ∣ x + 5 ∣ + \frac{1}{12} ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{x ^{2}}{x ^{4} - 20 x ^{2} - 125} d x

Take the partial fraction of

\frac{x ^{2}}{x ^{4} - 20 x ^{2} - 125} : \frac{1}{6 ( x ^{2} + 5 )} - \frac{1}{12 ( x + 5 )} + \frac{1}{12 ( x - 5 )}

= \int \frac{1}{6 ( x ^{2} + 5 )} - \frac{1}{12 ( x + 5 )} + \frac{1}{12 ( x - 5 )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{6 ( x ^{2} + 5 )} d x - \int \frac{1}{12 ( x + 5 )} d x + \int \frac{1}{12 ( x - 5 )} d x

\int \frac{1}{6 ( x ^{2} + 5 )} d x = \frac{1}{6 5} arctan (\frac{x}{5})

\int \frac{1}{12 ( x + 5 )} d x = \frac{1}{12} ln ∣ x + 5 ∣

\int \frac{1}{12 ( x - 5 )} d x = \frac{1}{12} ln ∣ x - 5 ∣

= \frac{1}{6 5} arctan (\frac{x}{5}) - \frac{1}{12} ln ∣ x + 5 ∣ + \frac{1}{12} ln ∣ x - 5 ∣

Add a constant to the solution

= \frac{1}{6 5} arctan (\frac{x}{5}) - \frac{1}{12} ln ∣ x + 5 ∣ + \frac{1}{12} ln ∣ x - 5 ∣ + C

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What is the integral of (x^2)/(x^4-20x^2-125) ?
The integral of (x^2)/(x^4-20x^2-125) is 1/(6sqrt(5))arctan(x/(sqrt(5)))-1/12 ln|x+5|+1/12 ln|x-5|+C