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

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

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

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

6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣ + C

Solution steps

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

Expand

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

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

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

\int \frac{6}{( x - 6 ) ( x + 4 )} d x = 6 (\frac{1}{10} ln ∣ x - 6 ∣ - \frac{1}{10} ln ∣ x + 4 ∣)

= 9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣) + 6 (\frac{1}{10} ln ∣ x - 6 ∣ - \frac{1}{10} ln ∣ x + 4 ∣)

Simplify

9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣) + 6 (\frac{1}{10} ln ∣ x - 6 ∣ - \frac{1}{10} ln ∣ x + 4 ∣) : 6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣

= 6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣

Add a constant to the solution

= 6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣ + C

\int_{1}^{2} \frac{x + 1}{x ( x ^{2} + 1 )} d x

y = ln (x^{2} + 4)

x \to 6 lim (x)

\frac{\partial}{\partial x} (\frac{2}{x} - x)

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