What is the integral of 1/(x^3+9x) ?

The integral of 1/(x^3+9x) is 1/9 ln|x|-1/18 ln|x^2+9|+C

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integral of 1/(x^3+9x)

\int \frac{1}{x ^{3} + 9 x} d x

\frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln x^{2} + 9 + C

Solution steps

\int \frac{1}{x ^{3} + 9 x} d x

Take the partial fraction of

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

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

\int \frac{1}{9 x} d x = \frac{1}{9} ln ∣ x ∣

\int \frac{x}{9 ( x ^{2} + 9 )} d x = \frac{1}{18} ln x^{2} + 9

= \frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln x^{2} + 9

Add a constant to the solution

= \frac{1}{9} ln ∣ x ∣ - \frac{1}{18} ln x^{2} + 9 + C

5 sec (x), at x = \frac{π}{3}

- x^{2} + 3 x, 2 x^{3} - x^{2} - 5 x

x \to 0 lim (\frac{( x - 1 )}{x})

\int \frac{y}{( y ^{2} + z ^{2} ) ^{\frac{3}{2}}} d y

(1, 7), (7, 3)

What is the integral of 1/(x^3+9x) ?
The integral of 1/(x^3+9x) is 1/9 ln|x|-1/18 ln|x^2+9|+C