What is the integral of (x^3+2x^2+5x+12)/(x+2) ?

The integral of (x^3+2x^2+5x+12)/(x+2) is -2x^2+x+1/3 (x+2)^3+2ln|x+2|+10+C

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integral of (x^3+2x^2+5x+12)/(x+2)

\int \frac{x ^{3} + 2 x ^{2} + 5 x + 12}{x + 2} d x

- 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} + 2 ln ∣ x + 2 ∣ + 10 + C

Solution steps

\int \frac{x ^{3} + 2 x ^{2} + 5 x + 12}{x + 2} d x

Expand

\frac{x ^{3} + 2 x ^{2} + 5 x + 12}{x + 2} : \frac{x ^{3}}{x + 2} + \frac{2 x ^{2}}{x + 2} + \frac{5 x}{x + 2} + \frac{12}{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{x ^{3}}{x + 2} d x + \int \frac{2 x ^{2}}{x + 2} d x + \int \frac{5 x}{x + 2} d x + \int \frac{12}{x + 2} d x

\int \frac{x ^{3}}{x + 2} d x = - 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12

\int \frac{2 x ^{2}}{x + 2} d x = x^{2} - 4 x + 8 ln ∣ x + 2 ∣ - 12

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

\int \frac{12}{x + 2} d x = 12 ln ∣ x + 2 ∣

= - 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12 + x^{2} - 4 x + 8 ln ∣ x + 2 ∣ - 12 + 5 (x + 2 - 2 ln ∣ x + 2 ∣) + 12 ln ∣ x + 2 ∣

Simplify

- 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12 + x^{2} - 4 x + 8 ln ∣ x + 2 ∣ - 12 + 5 (x + 2 - 2 ln ∣ x + 2 ∣) + 12 ln ∣ x + 2 ∣ : - 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} + 2 ln ∣ x + 2 ∣ + 10

= - 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} + 2 ln ∣ x + 2 ∣ + 10

Add a constant to the solution

= - 2 x^{2} + x + \frac{1}{3} (x + 2)^{3} + 2 ln ∣ x + 2 ∣ + 10 + C

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What is the integral of (x^3+2x^2+5x+12)/(x+2) ?
The integral of (x^3+2x^2+5x+12)/(x+2) is -2x^2+x+1/3 (x+2)^3+2ln|x+2|+10+C