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

The integral of (x^2+2x-3)/(x-2) is 6x+5ln|x-2|+1/2 (x-2)^2-12+C

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

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

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

Solution steps

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

Expand

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

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

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

Simplify

\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣ + 2 (x - 2 + 2 ln ∣ x - 2 ∣) - 3 ln ∣ x - 2 ∣ : 6 x + 5 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 12

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

Add a constant to the solution

= 6 x + 5 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 12 + C

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What is the integral of (x^2+2x-3)/(x-2) ?
The integral of (x^2+2x-3)/(x-2) is 6x+5ln|x-2|+1/2 (x-2)^2-12+C