What is the integral of (2x+5)/(x(x-2)) ?

The integral of (2x+5)/(x(x-2)) is -5/2 ln|x|+9/2 ln|x-2|+C

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integral of (2x+5)/(x(x-2))

\int \frac{2 x + 5}{x ( x - 2 )} d x

- \frac{5}{2} ln ∣ x ∣ + \frac{9}{2} ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{2 x + 5}{x ( x - 2 )} d x

Expand

\frac{2 x + 5}{x ( x - 2 )} : \frac{2 x}{x ( x - 2 )} + \frac{5}{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{2 x}{x ( x - 2 )} d x + \int \frac{5}{x ( x - 2 )} d x

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

\int \frac{5}{x ( x - 2 )} d x = - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

= 2 ln ∣ x - 2 ∣ - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Simplify

2 ln ∣ x - 2 ∣ - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) : - \frac{5}{2} ln ∣ x ∣ + \frac{9}{2} ln ∣ x - 2 ∣

= - \frac{5}{2} ln ∣ x ∣ + \frac{9}{2} ln ∣ x - 2 ∣

Add a constant to the solution

= - \frac{5}{2} ln ∣ x ∣ + \frac{9}{2} ln ∣ x - 2 ∣ + C

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What is the integral of (2x+5)/(x(x-2)) ?
The integral of (2x+5)/(x(x-2)) is -5/2 ln|x|+9/2 ln|x-2|+C