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

The integral of (2x^2+x+6)/(x^2+x-2) is 2x+ln|x-1|-2ln|x+2|-2ln| 2/3 x+4/3 |+2ln| 2/3 x-2/3 |+C

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

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

2 x + ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ - 2 ln \frac{2}{3} x + \frac{4}{3} + 2 ln \frac{2}{3} x - \frac{2}{3} + C

Solution steps

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

Expand

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

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

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

\int \frac{6}{x ^{2} + x - 2} d x = - 4 (\frac{1}{2} ln \frac{4}{3} + \frac{2}{3} x - \frac{1}{2} ln - \frac{2}{3} + \frac{2}{3} x)

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

Simplify

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

= 2 x + ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ - 2 ln \frac{2}{3} x + \frac{4}{3} + 2 ln \frac{2}{3} x - \frac{2}{3}

Add a constant to the solution

= 2 x + ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ - 2 ln \frac{2}{3} x + \frac{4}{3} + 2 ln \frac{2}{3} x - \frac{2}{3} + C

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What is the integral of (2x^2+x+6)/(x^2+x-2) ?
The integral of (2x^2+x+6)/(x^2+x-2) is 2x+ln|x-1|-2ln|x+2|-2ln| 2/3 x+4/3 |+2ln| 2/3 x-2/3 |+C