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

The integral of ((x+2)^2)/(x-2) is (x^2)/2+6x+16ln|x-2|+C

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

\int \frac{( x + 2 ) ^{2}}{x - 2} d x

\frac{x ^{2}}{2} + 6 x + 16 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{( x + 2 ) ^{2}}{x - 2} d x

Long division

\frac{( x + 2 ) ^{2}}{x - 2} : x + 6 + \frac{16}{x - 2}

= \int x + 6 + \frac{16}{x - 2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x d x + \int 6 d x + \int \frac{16}{x - 2} d x

\int x d x = \frac{x ^{2}}{2}

\int 6 d x = 6 x

\int \frac{16}{x - 2} d x = 16 ln ∣ x - 2 ∣

= \frac{x ^{2}}{2} + 6 x + 16 ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 6 x + 16 ln ∣ x - 2 ∣ + C

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