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

The integral of (x^2+x+6)/(x-2) is 5x+12ln|x-2|+1/2 (x-2)^2-10+C

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

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

5 x + 12 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 10 + C

Solution steps

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

Expand

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

\int \frac{6}{x - 2} d x = 6 ln ∣ x - 2 ∣

= \frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣ + x - 2 + 2 ln ∣ x - 2 ∣ + 6 ln ∣ x - 2 ∣

Simplify

\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣ + x - 2 + 2 ln ∣ x - 2 ∣ + 6 ln ∣ x - 2 ∣ : 5 x + 12 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 10

= 5 x + 12 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 10

Add a constant to the solution

= 5 x + 12 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 10 + C

What is the integral of (x^2+x+6)/(x-2) ?
The integral of (x^2+x+6)/(x-2) is 5x+12ln|x-2|+1/2 (x-2)^2-10+C