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

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

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

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

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

Solution steps

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

Long division

\frac{x ^{3} + 3 x}{( 2 x - 2 ) ( x + 2 )} : \frac{x}{2} - \frac{1}{2} + \frac{3 x - 1}{x ^{2} + x - 2}

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

\int \frac{1}{2} d x = \frac{1}{2} x

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

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

Add a constant to the solution

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

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