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

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

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

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

x - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + C

Solution steps

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

Long division

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

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

\int 1 d x = x

\int \frac{3 x - 2}{x ^{2} - 3 x + 2} d x = - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣

= x - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣

Add a constant to the solution

= x - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + C

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