What is the integral of (x^2+3x-2)/(x(x+1)(x-2)) ?

The integral of (x^2+3x-2)/(x(x+1)(x-2)) is -4/3 ln|x+1|+4/3 ln|x-2|+ln|x|+C

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integral of (x^2+3x-2)/(x(x+1)(x-2))

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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x e^{- x}

What is the integral of (x^2+3x-2)/(x(x+1)(x-2)) ?
The integral of (x^2+3x-2)/(x(x+1)(x-2)) is -4/3 ln|x+1|+4/3 ln|x-2|+ln|x|+C