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

The integral of (x^3-x^2+x+1)/(x+2) is -3x^2+5x+1/3 (x+2)^3-13ln|x+2|-1/2 (x+2)^2+22+C

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

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

- 3 x^{2} + 5 x + \frac{1}{3} (x + 2)^{3} - 13 ln ∣ x + 2 ∣ - \frac{1}{2} (x + 2)^{2} + 22 + C

Solution steps

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

Expand

\frac{x ^{3} - x ^{2} + x + 1}{x + 2} : \frac{x ^{3}}{x + 2} - \frac{x ^{2}}{x + 2} + \frac{x}{x + 2} + \frac{1}{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{x ^{2}}{x + 2} d x + \int \frac{x}{x + 2} d x + \int \frac{1}{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{x ^{2}}{x + 2} d x = \frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 4 ln ∣ x + 2 ∣

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

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

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

Simplify

- 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12 - (\frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 4 ln ∣ x + 2 ∣) + x + 2 - 2 ln ∣ x + 2 ∣ + ln ∣ x + 2 ∣ : - 3 x^{2} + 5 x + \frac{1}{3} (x + 2)^{3} - 13 ln ∣ x + 2 ∣ - \frac{1}{2} (x + 2)^{2} + 22

= - 3 x^{2} + 5 x + \frac{1}{3} (x + 2)^{3} - 13 ln ∣ x + 2 ∣ - \frac{1}{2} (x + 2)^{2} + 22

Add a constant to the solution

= - 3 x^{2} + 5 x + \frac{1}{3} (x + 2)^{3} - 13 ln ∣ x + 2 ∣ - \frac{1}{2} (x + 2)^{2} + 22 + C

\int \frac{x}{x ^{2} - a ^{2}} d x

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\int sin (5 x) cos^{3} (5 x) d x

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

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

What is the integral of (x^3-x^2+x+1)/(x+2) ?
The integral of (x^3-x^2+x+1)/(x+2) is -3x^2+5x+1/3 (x+2)^3-13ln|x+2|-1/2 (x+2)^2+22+C