What is the integral of (x^2+9x+2)/((x-1)(x+1)(x+2)) ?

The integral of (x^2+9x+2)/((x-1)(x+1)(x+2)) is

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integral of (x^2+9x+2)/((x-1)(x+1)(x+2))

\int \frac{x ^{2} + 9 x + 2}{( x - 1 ) ( x + 1 ) ( x + 2 )} d x

\frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ + \frac{7}{3} ln ∣ x - 1 ∣ - \frac{7}{3} ln ∣ x + 2 ∣ + 3 ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{x ^{2} + 9 x + 2}{( x - 1 ) ( x + 1 ) ( x + 2 )} d x

Apply u-substitution

: \int \frac{u ^{2} + 7 u - 6}{u ( u - 2 ) ( u + 1 )} d u

= \int \frac{u ^{2} + 7 u - 6}{u ( u - 2 ) ( u + 1 )} d u

Expand

\frac{u ^{2} + 7 u - 6}{u ( u - 2 ) ( u + 1 )} : \frac{u ^{2}}{u ( u - 2 ) ( u + 1 )} + \frac{7 u}{u ( u - 2 ) ( u + 1 )} - \frac{6}{u ( u - 2 ) ( u + 1 )}

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{u ^{2}}{u ( u - 2 ) ( u + 1 )} d u + \int \frac{7 u}{u ( u - 2 ) ( u + 1 )} d u - \int \frac{6}{u ( u - 2 ) ( u + 1 )} d u

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

\int \frac{7 u}{u ( u - 2 ) ( u + 1 )} d u = 7 (\frac{1}{3} ln ∣ u - 2 ∣ - \frac{1}{3} ln ∣ u + 1 ∣)

\int \frac{6}{u ( u - 2 ) ( u + 1 )} d u = - 3 ln ∣ u ∣ + ln ∣ u - 2 ∣ + 2 ln ∣ u + 1 ∣

= \frac{2}{3} ln ∣ u - 2 ∣ + \frac{1}{3} ln ∣ u + 1 ∣ + 7 (\frac{1}{3} ln ∣ u - 2 ∣ - \frac{1}{3} ln ∣ u + 1 ∣) - (- 3 ln ∣ u ∣ + ln ∣ u - 2 ∣ + 2 ln ∣ u + 1 ∣)

Substitute back

u = x + 1

= \frac{2}{3} ln ∣ x + 1 - 2 ∣ + \frac{1}{3} ln ∣ x + 1 + 1 ∣ + 7 (\frac{1}{3} ln ∣ x + 1 - 2 ∣ - \frac{1}{3} ln ∣ x + 1 + 1 ∣) - (- 3 ln ∣ x + 1 ∣ + ln ∣ x + 1 - 2 ∣ + 2 ln ∣ x + 1 + 1 ∣)

Simplify

\frac{2}{3} ln ∣ x + 1 - 2 ∣ + \frac{1}{3} ln ∣ x + 1 + 1 ∣ + 7 (\frac{1}{3} ln ∣ x + 1 - 2 ∣ - \frac{1}{3} ln ∣ x + 1 + 1 ∣) - (- 3 ln ∣ x + 1 ∣ + ln ∣ x + 1 - 2 ∣ + 2 ln ∣ x + 1 + 1 ∣) : \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ + \frac{7}{3} ln ∣ x - 1 ∣ - \frac{7}{3} ln ∣ x + 2 ∣ + 3 ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣

= \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ + \frac{7}{3} ln ∣ x - 1 ∣ - \frac{7}{3} ln ∣ x + 2 ∣ + 3 ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ + \frac{7}{3} ln ∣ x - 1 ∣ - \frac{7}{3} ln ∣ x + 2 ∣ + 3 ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ + C

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