What is the integral of (x-18)/(4x^3+9x) ?

The integral of (x-18)/(4x^3+9x) is -2ln|x|+ln|4x^2+9|+1/6 arctan(2/3 x)+C

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integral of (x-18)/(4x^3+9x)

\int \frac{x - 18}{4 x ^{3} + 9 x} d x

- 2 ln ∣ x ∣ + ln 4 x^{2} + 9 + \frac{1}{6} arctan (\frac{2}{3} x) + C

Solution steps

\int \frac{x - 18}{4 x ^{3} + 9 x} d x

Take the partial fraction of

\frac{x - 18}{4 x ^{3} + 9 x} : - \frac{2}{x} + \frac{8 x + 1}{4 x ^{2} + 9}

= \int - \frac{2}{x} + \frac{8 x + 1}{4 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{2}{x} d x + \int \frac{8 x + 1}{4 x ^{2} + 9} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{8 x + 1}{4 x ^{2} + 9} d x = ln 4 x^{2} + 9 + \frac{1}{6} arctan (\frac{2}{3} x)

= - 2 ln ∣ x ∣ + ln 4 x^{2} + 9 + \frac{1}{6} arctan (\frac{2}{3} x)

Add a constant to the solution

= - 2 ln ∣ x ∣ + ln 4 x^{2} + 9 + \frac{1}{6} arctan (\frac{2}{3} x) + C

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What is the integral of (x-18)/(4x^3+9x) ?
The integral of (x-18)/(4x^3+9x) is -2ln|x|+ln|4x^2+9|+1/6 arctan(2/3 x)+C