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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

Add a constant to the solution

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

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