What is the integral of (2x^3+3x-4)/(x+2) ?

The integral of (2x^3+3x-4)/(x+2) is -6x^2+3x+2/3 (x+2)^3-26ln|x+2|+30+C

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integral of (2x^3+3x-4)/(x+2)

\int \frac{2 x ^{3} + 3 x - 4}{x + 2} d x

- 6 x^{2} + 3 x + \frac{2}{3} (x + 2)^{3} - 26 ln ∣ x + 2 ∣ + 30 + C

Solution steps

\int \frac{2 x ^{3} + 3 x - 4}{x + 2} d x

Expand

\frac{2 x ^{3} + 3 x - 4}{x + 2} : \frac{2 x ^{3}}{x + 2} + \frac{3 x}{x + 2} - \frac{4}{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{2 x ^{3}}{x + 2} d x + \int \frac{3 x}{x + 2} d x - \int \frac{4}{x + 2} d x

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

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

\int \frac{4}{x + 2} d x = 4 ln ∣ x + 2 ∣

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

Simplify

2 (- 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12) + 3 (x + 2 - 2 ln ∣ x + 2 ∣) - 4 ln ∣ x + 2 ∣ : - 6 x^{2} + 3 x + \frac{2}{3} (x + 2)^{3} - 26 ln ∣ x + 2 ∣ + 30

= - 6 x^{2} + 3 x + \frac{2}{3} (x + 2)^{3} - 26 ln ∣ x + 2 ∣ + 30

Add a constant to the solution

= - 6 x^{2} + 3 x + \frac{2}{3} (x + 2)^{3} - 26 ln ∣ x + 2 ∣ + 30 + C

\int \frac{x ^{2} + 1}{x ^{2} - 4 x + 4} d x

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What is the integral of (2x^3+3x-4)/(x+2) ?
The integral of (2x^3+3x-4)/(x+2) is -6x^2+3x+2/3 (x+2)^3-26ln|x+2|+30+C