What is the integral of (9x)/((x-6)(x+4)) ?

The integral of (9x)/((x-6)(x+4)) is 9(3/5 ln|x-6|+2/5 ln|x+4|)+C

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integral of (9x)/((x-6)(x+4))

\int \frac{9 x}{( x - 6 ) ( x + 4 )} d x

9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣) + C

Solution steps

\int \frac{9 x}{( x - 6 ) ( x + 4 )} d x

Take the constant out:

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

= 9 \cdot \int \frac{x}{( x - 6 ) ( x + 4 )} d x

Take the partial fraction of

\frac{x}{( x - 6 ) ( x + 4 )} : \frac{3}{5 ( x - 6 )} + \frac{2}{5 ( x + 4 )}

= 9 \cdot \int \frac{3}{5 ( x - 6 )} + \frac{2}{5 ( x + 4 )} 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

= 9 (\int \frac{3}{5 ( x - 6 )} d x + \int \frac{2}{5 ( x + 4 )} d x)

\int \frac{3}{5 ( x - 6 )} d x = \frac{3}{5} ln ∣ x - 6 ∣

\int \frac{2}{5 ( x + 4 )} d x = \frac{2}{5} ln ∣ x + 4 ∣

= 9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣)

Add a constant to the solution

= 9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣) + C

What is the integral of (9x)/((x-6)(x+4)) ?
The integral of (9x)/((x-6)(x+4)) is 9(3/5 ln|x-6|+2/5 ln|x+4|)+C