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

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

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integral of (x+3)/(x(x+2))

\int \frac{x + 3}{x ( x + 2 )} d x

\frac{3}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{x + 3}{x ( x + 2 )} d x

Expand

\frac{x + 3}{x ( x + 2 )} : \frac{x}{x ( x + 2 )} + \frac{3}{x ( x + 2 )}

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 + 2 )} d x + \int \frac{3}{x ( x + 2 )} d x

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

\int \frac{3}{x ( x + 2 )} d x = 3 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣)

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

Simplify

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

= \frac{3}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{3}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣ + C

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