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

The integral of (x+3)/((x+1)(x-2)) is -2/3 ln|x+1|+5/3 ln|x-2|+C

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

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

- \frac{2}{3} ln ∣ x + 1 ∣ + \frac{5}{3} ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x + 3}{( x + 1 ) ( x - 2 )} : - \frac{2}{3 ( x + 1 )} + \frac{5}{3 ( x - 2 )}

= \int - \frac{2}{3 ( x + 1 )} + \frac{5}{3 ( 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{2}{3 ( x + 1 )} d x + \int \frac{5}{3 ( x - 2 )} d x

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

\int \frac{5}{3 ( x - 2 )} d x = \frac{5}{3} ln ∣ x - 2 ∣

= - \frac{2}{3} ln ∣ x + 1 ∣ + \frac{5}{3} ln ∣ x - 2 ∣

Add a constant to the solution

= - \frac{2}{3} ln ∣ x + 1 ∣ + \frac{5}{3} ln ∣ x - 2 ∣ + C

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What is the integral of (x+3)/((x+1)(x-2)) ?
The integral of (x+3)/((x+1)(x-2)) is -2/3 ln|x+1|+5/3 ln|x-2|+C