What is the integral of (x^5+9x^3-9x^2-9)/(x^3+9x) ?

The integral of (x^5+9x^3-9x^2-9)/(x^3+9x) is (x^3)/3-4ln|x^2+9|-ln|x|+C

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integral of (x^5+9x^3-9x^2-9)/(x^3+9x)

\int \frac{x ^{5} + 9 x ^{3} - 9 x ^{2} - 9}{x ^{3} + 9 x} d x

\frac{x ^{3}}{3} - 4 ln x^{2} + 9 - ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{5} + 9 x ^{3} - 9 x ^{2} - 9}{x ^{3} + 9 x} d x

Long division

\frac{x ^{5} + 9 x ^{3} - 9 x ^{2} - 9}{x ^{3} + 9 x} : x^{2} + \frac{- 9 x ^{2} - 9}{x ^{3} + 9 x}

= \int x^{2} + \frac{- 9 x ^{2} - 9}{x ^{3} + 9 x} 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 x^{2} d x + \int \frac{- 9 x ^{2} - 9}{x ^{3} + 9 x} d x

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

\int \frac{- 9 x ^{2} - 9}{x ^{3} + 9 x} d x = - 4 ln x^{2} + 9 - ln ∣ x ∣

= \frac{x ^{3}}{3} - 4 ln x^{2} + 9 - ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{3}}{3} - 4 ln x^{2} + 9 - ln ∣ x ∣ + C

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What is the integral of (x^5+9x^3-9x^2-9)/(x^3+9x) ?
The integral of (x^5+9x^3-9x^2-9)/(x^3+9x) is (x^3)/3-4ln|x^2+9|-ln|x|+C