What is the integral of (x^2-6)/(x(x-3)) ?

The integral of (x^2-6)/(x(x-3)) is x+ln|x-3|+2ln|x|-3+C

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integral of (x^2-6)/(x(x-3))

\int \frac{x ^{2} - 6}{x ( x - 3 )} d x

x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3 + C

Solution steps

\int \frac{x ^{2} - 6}{x ( x - 3 )} d x

Expand

\frac{x ^{2} - 6}{x ( x - 3 )} : \frac{x ^{2}}{x ( x - 3 )} - \frac{6}{x ( x - 3 )}

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{x ^{2}}{x ( x - 3 )} d x - \int \frac{6}{x ( x - 3 )} d x

\int \frac{x ^{2}}{x ( x - 3 )} d x = x - 3 + 3 ln ∣ x - 3 ∣

\int \frac{6}{x ( x - 3 )} d x = 6 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 3 ∣)

= x - 3 + 3 ln ∣ x - 3 ∣ - 6 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 3 ∣)

Simplify

x - 3 + 3 ln ∣ x - 3 ∣ - 6 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 3 ∣) : x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3

= x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3

Add a constant to the solution

= x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3 + C

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