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

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

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

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

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

Solution steps

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

Expand

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

\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 - 1 + ln ∣ x - 1 ∣) + 2 ln ∣ x - 1 ∣

Simplify

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

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

Add a constant to the solution

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

e^{- 2 t} cos^{2} (t)

\frac{\partial}{\partial y} (24 y^{2} - 12 x)

\frac{\partial}{\partial x} (7 Y (x))

\frac{- 5}{s ( 6 ( 5 s + 1 ) )}

f (x) = (7 x^{6} + 4 x^{3})^{4}

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