What is the integral of (3x^3-x+5)/(x+2) ?

The integral of (3x^3-x+5)/(x+2) is -9x^2-x+(x+2)^3-17ln|x+2|+34+C

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integral of (3x^3-x+5)/(x+2)

\int \frac{3 x ^{3} - x + 5}{x + 2} d x

- 9 x^{2} - x + (x + 2)^{3} - 17 ln ∣ x + 2 ∣ + 34 + C

Solution steps

\int \frac{3 x ^{3} - x + 5}{x + 2} d x

Expand

\frac{3 x ^{3} - x + 5}{x + 2} : \frac{3 x ^{3}}{x + 2} - \frac{x}{x + 2} + \frac{5}{x + 2}

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{3 x ^{3}}{x + 2} d x - \int \frac{x}{x + 2} d x + \int \frac{5}{x + 2} d x

\int \frac{3 x ^{3}}{x + 2} d x = - 9 x^{2} + (x + 2)^{3} - 24 ln ∣ x + 2 ∣ + 36

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

\int \frac{5}{x + 2} d x = 5 ln ∣ x + 2 ∣

= - 9 x^{2} + (x + 2)^{3} - 24 ln ∣ x + 2 ∣ + 36 - (x + 2 - 2 ln ∣ x + 2 ∣) + 5 ln ∣ x + 2 ∣

Simplify

- 9 x^{2} + (x + 2)^{3} - 24 ln ∣ x + 2 ∣ + 36 - (x + 2 - 2 ln ∣ x + 2 ∣) + 5 ln ∣ x + 2 ∣ : - 9 x^{2} - x + (x + 2)^{3} - 17 ln ∣ x + 2 ∣ + 34

= - 9 x^{2} - x + (x + 2)^{3} - 17 ln ∣ x + 2 ∣ + 34

Add a constant to the solution

= - 9 x^{2} - x + (x + 2)^{3} - 17 ln ∣ x + 2 ∣ + 34 + C

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What is the integral of (3x^3-x+5)/(x+2) ?
The integral of (3x^3-x+5)/(x+2) is -9x^2-x+(x+2)^3-17ln|x+2|+34+C