What is the integral of (x^3+6x^2+7x+2)/(x^2-x-2) ?

The integral of (x^3+6x^2+7x+2)/(x^2-x-2) is 9x+16ln|x-2|+1/2 (x-2)^2-18+C

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integral of (x^3+6x^2+7x+2)/(x^2-x-2)

\int \frac{x ^{3} + 6 x ^{2} + 7 x + 2}{x ^{2} - x - 2} d x

9 x + 16 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 18 + C

Solution steps

\int \frac{x ^{3} + 6 x ^{2} + 7 x + 2}{x ^{2} - x - 2} d x

Simplify

\frac{x ^{3} + 6 x ^{2} + 7 x + 2}{x ^{2} - x - 2} : \frac{x ^{2} + 5 x + 2}{x - 2}

= \int \frac{x ^{2} + 5 x + 2}{x - 2} d x

Expand

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

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

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

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

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

Simplify

\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣ + 5 (x - 2 + 2 ln ∣ x - 2 ∣) + 2 ln ∣ x - 2 ∣ : 9 x + 16 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 18

= 9 x + 16 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 18

Add a constant to the solution

= 9 x + 16 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 18 + C

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What is the integral of (x^3+6x^2+7x+2)/(x^2-x-2) ?
The integral of (x^3+6x^2+7x+2)/(x^2-x-2) is 9x+16ln|x-2|+1/2 (x-2)^2-18+C