What is the integral of (x^2+3)/(x^2(x^2+9)) ?

The integral of (x^2+3)/(x^2(x^2+9)) is -1/(3x)+2/9 arctan(x/3)+C

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integral of (x^2+3)/(x^2(x^2+9))

\int \frac{x ^{2} + 3}{x ^{2} ( x ^{2} + 9 )} d x

- \frac{1}{3 x} + \frac{2}{9} arctan (\frac{x}{3}) + C

Solution steps

\int \frac{x ^{2} + 3}{x ^{2} ( x ^{2} + 9 )} d x

Take the partial fraction of

\frac{x ^{2} + 3}{x ^{2} ( x ^{2} + 9 )} : \frac{1}{3 x ^{2}} + \frac{2}{3 ( x ^{2} + 9 )}

= \int \frac{1}{3 x ^{2}} + \frac{2}{3 ( x ^{2} + 9 )} 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}{3 x ^{2}} d x + \int \frac{2}{3 ( x ^{2} + 9 )} d x

\int \frac{1}{3 x ^{2}} d x = - \frac{1}{3 x}

\int \frac{2}{3 ( x ^{2} + 9 )} d x = \frac{2}{9} arctan (\frac{x}{3})

= - \frac{1}{3 x} + \frac{2}{9} arctan (\frac{x}{3})

Add a constant to the solution

= - \frac{1}{3 x} + \frac{2}{9} arctan (\frac{x}{3}) + C

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What is the integral of (x^2+3)/(x^2(x^2+9)) ?
The integral of (x^2+3)/(x^2(x^2+9)) is -1/(3x)+2/9 arctan(x/3)+C