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

The integral of (x^3-1)/(x+2) is (x^3)/3-x^2+4x-9ln|x+2|+C

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

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

\frac{x ^{3}}{3} - x^{2} + 4 x - 9 ln ∣ x + 2 ∣ + C

Solution steps

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

Long division

\frac{x ^{3} - 1}{x + 2} : x^{2} - 2 x + 4 - \frac{9}{x + 2}

= \int x^{2} - 2 x + 4 - \frac{9}{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 2 x d x + \int 4 d x - \int \frac{9}{x + 2} d x

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

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

\int 4 d x = 4 x

\int \frac{9}{x + 2} d x = 9 ln ∣ x + 2 ∣

= \frac{x ^{3}}{3} - x^{2} + 4 x - 9 ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{x ^{3}}{3} - x^{2} + 4 x - 9 ln ∣ x + 2 ∣ + C

\frac{d}{d x} (sin (5 x))

x \to - 2 lim (4 x^{2 - 5 x + 2})

f (x) = 1 + 2 e^{x}, at x = 1

\int 4 x ln (x^{11}) d x

(- 7, - 2), (- 6, 7)

What is the integral of (x^3-1)/(x+2) ?
The integral of (x^3-1)/(x+2) is (x^3)/3-x^2+4x-9ln|x+2|+C