What is the integral of (x+8)/(x(x+6)) ?

The integral of (x+8)/(x(x+6)) is 4/3 ln|x|-1/3 ln|x+6|+C

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integral of (x+8)/(x(x+6))

\int \frac{x + 8}{x ( x + 6 )} d x

\frac{4}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 6 ∣ + C

Solution steps

\int \frac{x + 8}{x ( x + 6 )} d x

Expand

\frac{x + 8}{x ( x + 6 )} : \frac{x}{x ( x + 6 )} + \frac{8}{x ( x + 6 )}

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{x}{x ( x + 6 )} d x + \int \frac{8}{x ( x + 6 )} d x

\int \frac{x}{x ( x + 6 )} d x = ln ∣ x + 6 ∣

\int \frac{8}{x ( x + 6 )} d x = 8 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x + 6 ∣)

= ln ∣ x + 6 ∣ + 8 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x + 6 ∣)

Simplify

ln ∣ x + 6 ∣ + 8 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x + 6 ∣) : \frac{4}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 6 ∣

= \frac{4}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 6 ∣

Add a constant to the solution

= \frac{4}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 6 ∣ + C

h \to 0 lim (\frac{sinh ( x )}{h})

\frac{s + 2}{s ^{2} + 2 s + 1}

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\int (cos^{4} (x) - cos^{6} (x)) d x

What is the integral of (x+8)/(x(x+6)) ?
The integral of (x+8)/(x(x+6)) is 4/3 ln|x|-1/3 ln|x+6|+C