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

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

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

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

\frac{x ^{3}}{3} + ln ∣ x + 2 ∣ + C

Solution steps

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

Long division

\frac{x ^{3} + 2 x ^{2} + 1}{x + 2} : x^{2} + \frac{1}{x + 2}

= \int x^{2} + \frac{1}{x + 2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} d x + \int \frac{1}{x + 2} d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

= \frac{x ^{3}}{3} + ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{x ^{3}}{3} + ln ∣ x + 2 ∣ + C

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