What is the integral of 1/(x(x-5)) ?

The integral of 1/(x(x-5)) is -1/5 ln|x|+1/5 ln|x-5|+C

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integral of 1/(x(x-5))

\int \frac{1}{x ( x - 5 )} d x

- \frac{1}{5} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{1}{x ( x - 5 )} d x

Take the partial fraction of

\frac{1}{x ( x - 5 )} : - \frac{1}{5 x} + \frac{1}{5 ( x - 5 )}

= \int - \frac{1}{5 x} + \frac{1}{5 ( x - 5 )} d 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{1}{5 x} d x + \int \frac{1}{5 ( x - 5 )} d x

\int \frac{1}{5 x} d x = \frac{1}{5} ln ∣ x ∣

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

= - \frac{1}{5} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 5 ∣

Add a constant to the solution

= - \frac{1}{5} ln ∣ x ∣ + \frac{1}{5} ln ∣ x - 5 ∣ + C

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What is the integral of 1/(x(x-5)) ?
The integral of 1/(x(x-5)) is -1/5 ln|x|+1/5 ln|x-5|+C