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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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