What is the integral of ((x-1)(x+5))/(x+2) ?

The integral of ((x-1)(x+5))/(x+2) is (x^2)/2+2x-9ln|x+2|+C

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integral of ((x-1)(x+5))/(x+2)

\int \frac{( x - 1 ) ( x + 5 )}{x + 2} d x

\frac{x ^{2}}{2} + 2 x - 9 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{( x - 1 ) ( x + 5 )}{x + 2} d x

Long division

\frac{( x - 1 ) ( x + 5 )}{x + 2} : x + 2 - \frac{9}{x + 2}

= \int x + 2 - \frac{9}{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 2 d x - \int \frac{9}{x + 2} d x

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

\int 2 d x = 2 x

\int \frac{9}{x + 2} d x = 9 ln ∣ x + 2 ∣

= \frac{x ^{2}}{2} + 2 x - 9 ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 2 x - 9 ln ∣ x + 2 ∣ + C

What is the integral of ((x-1)(x+5))/(x+2) ?
The integral of ((x-1)(x+5))/(x+2) is (x^2)/2+2x-9ln|x+2|+C