What is the integral of (4x)/((x-3)(x+2)) ?

The integral of (4x)/((x-3)(x+2)) is 4(3/5 ln|x-3|+2/5 ln|x+2|)+C

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integral of (4x)/((x-3)(x+2))

\int \frac{4 x}{( x - 3 ) ( x + 2 )} d x

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

Solution steps

\int \frac{4 x}{( x - 3 ) ( x + 2 )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{x}{( x - 3 ) ( x + 2 )} d x

Take the partial fraction of

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

= 4 \cdot \int \frac{3}{5 ( x - 3 )} + \frac{2}{5 ( 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

= 4 (\int \frac{3}{5 ( x - 3 )} d x + \int \frac{2}{5 ( x + 2 )} d x)

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

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

= 4 (\frac{3}{5} ln ∣ x - 3 ∣ + \frac{2}{5} ln ∣ x + 2 ∣)

Add a constant to the solution

= 4 (\frac{3}{5} ln ∣ x - 3 ∣ + \frac{2}{5} ln ∣ x + 2 ∣) + C

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What is the integral of (4x)/((x-3)(x+2)) ?
The integral of (4x)/((x-3)(x+2)) is 4(3/5 ln|x-3|+2/5 ln|x+2|)+C