What is the integral of (3x-1)/((x-1)^2(x-3)) ?

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

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integral of (3x-1)/((x-1)^2(x-3))

\int \frac{3 x - 1}{( x - 1 ) ^{2} ( x - 3 )} d x

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

Solution steps

\int \frac{3 x - 1}{( x - 1 ) ^{2} ( x - 3 )} d x

Take the partial fraction of

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

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

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