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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

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