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

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

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

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

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

Solution steps

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

Expand

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

Simplify

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

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

Add a constant to the solution

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

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