What is the integral of (x-5)/(x^2(x-1)) ?

The integral of (x-5)/(x^2(x-1)) is 4ln|x|-5/x-4ln|x-1|+C

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integral of (x-5)/(x^2(x-1))

\int \frac{x - 5}{x ^{2} ( x - 1 )} d x

4 ln ∣ x ∣ - \frac{5}{x} - 4 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{x - 5}{x ^{2} ( x - 1 )} d x

Take the partial fraction of

\frac{x - 5}{x ^{2} ( x - 1 )} : \frac{4}{x} + \frac{5}{x ^{2}} - \frac{4}{x - 1}

= \int \frac{4}{x} + \frac{5}{x ^{2}} - \frac{4}{x - 1} 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{4}{x} d x + \int \frac{5}{x ^{2}} d x - \int \frac{4}{x - 1} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int \frac{5}{x ^{2}} d x = - \frac{5}{x}

\int \frac{4}{x - 1} d x = 4 ln ∣ x - 1 ∣

= 4 ln ∣ x ∣ - \frac{5}{x} - 4 ln ∣ x - 1 ∣

Add a constant to the solution

= 4 ln ∣ x ∣ - \frac{5}{x} - 4 ln ∣ x - 1 ∣ + C

What is the integral of (x-5)/(x^2(x-1)) ?
The integral of (x-5)/(x^2(x-1)) is 4ln|x|-5/x-4ln|x-1|+C