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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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