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

The integral of (x-1)/(x(x+3)) is -1/3 ln|x|+4/3 ln|x+3|+C

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integral of (x-1)/(x(x+3))

\int \frac{x - 1}{x ( x + 3 )} d x

- \frac{1}{3} ln ∣ x ∣ + \frac{4}{3} ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{x - 1}{x ( x + 3 )} d x

Expand

\frac{x - 1}{x ( x + 3 )} : \frac{x}{x ( x + 3 )} - \frac{1}{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}{x ( x + 3 )} d x - \int \frac{1}{x ( x + 3 )} d x

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

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

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

Simplify

ln ∣ x + 3 ∣ - (\frac{1}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 3 ∣) : - \frac{1}{3} ln ∣ x ∣ + \frac{4}{3} ln ∣ x + 3 ∣

= - \frac{1}{3} ln ∣ x ∣ + \frac{4}{3} ln ∣ x + 3 ∣

Add a constant to the solution

= - \frac{1}{3} ln ∣ x ∣ + \frac{4}{3} ln ∣ x + 3 ∣ + C

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What is the integral of (x-1)/(x(x+3)) ?
The integral of (x-1)/(x(x+3)) is -1/3 ln|x|+4/3 ln|x+3|+C