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

The integral of (x^3+2x-1)/(x-1) is 5x+2ln|x-1|+1/3 (x-1)^3+3/2 (x-1)^2-5+C

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

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

5 x + 2 ln ∣ x - 1 ∣ + \frac{1}{3} (x - 1)^{3} + \frac{3}{2} (x - 1)^{2} - 5 + C

Solution steps

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

Expand

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

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

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

Simplify

\frac{( x - 1 ) ^{3}}{3} + \frac{3 ( x - 1 ) ^{2}}{2} + 3 (x - 1) + ln ∣ x - 1 ∣ + 2 (x - 1 + ln ∣ x - 1 ∣) - ln ∣ x - 1 ∣ : 5 x + 2 ln ∣ x - 1 ∣ + \frac{1}{3} (x - 1)^{3} + \frac{3}{2} (x - 1)^{2} - 5

= 5 x + 2 ln ∣ x - 1 ∣ + \frac{1}{3} (x - 1)^{3} + \frac{3}{2} (x - 1)^{2} - 5

Add a constant to the solution

= 5 x + 2 ln ∣ x - 1 ∣ + \frac{1}{3} (x - 1)^{3} + \frac{3}{2} (x - 1)^{2} - 5 + C

What is the integral of (x^3+2x-1)/(x-1) ?
The integral of (x^3+2x-1)/(x-1) is 5x+2ln|x-1|+1/3 (x-1)^3+3/2 (x-1)^2-5+C