What is the integral of 1/(3x^2+27x) ?

The integral of 1/(3x^2+27x) is 1/27 ln|x|-1/27 ln|x+9|+C

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integral of 1/(3x^2+27x)

\int \frac{1}{3 x ^{2} + 27 x} d x

\frac{1}{27} ln ∣ x ∣ - \frac{1}{27} ln ∣ x + 9 ∣ + C

Solution steps

\int \frac{1}{3 x ^{2} + 27 x} d x

Take the partial fraction of

\frac{1}{3 x ^{2} + 27 x} : \frac{1}{27 x} - \frac{1}{27 ( x + 9 )}

= \int \frac{1}{27 x} - \frac{1}{27 ( x + 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}{27 x} d x - \int \frac{1}{27 ( x + 9 )} d x

\int \frac{1}{27 x} d x = \frac{1}{27} ln ∣ x ∣

\int \frac{1}{27 ( x + 9 )} d x = \frac{1}{27} ln ∣ x + 9 ∣

= \frac{1}{27} ln ∣ x ∣ - \frac{1}{27} ln ∣ x + 9 ∣

Add a constant to the solution

= \frac{1}{27} ln ∣ x ∣ - \frac{1}{27} ln ∣ x + 9 ∣ + C

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What is the integral of 1/(3x^2+27x) ?
The integral of 1/(3x^2+27x) is 1/27 ln|x|-1/27 ln|x+9|+C