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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

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