What is the integral of 8/(x(x+2)) ?

The integral of 8/(x(x+2)) is 8(1/2 ln|x|-1/2 ln|x+2|)+C

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integral of 8/(x(x+2))

\int \frac{8}{x ( x + 2 )} d x

8 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣) + C

Solution steps

\int \frac{8}{x ( x + 2 )} d x

Take the constant out:

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

= 8 \cdot \int \frac{1}{x ( x + 2 )} d x

Take the partial fraction of

\frac{1}{x ( x + 2 )} : \frac{1}{2 x} - \frac{1}{2 ( x + 2 )}

= 8 \cdot \int \frac{1}{2 x} - \frac{1}{2 ( 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

= 8 (\int \frac{1}{2 x} d x - \int \frac{1}{2 ( x + 2 )} d x)

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

\int \frac{1}{2 ( x + 2 )} d x = \frac{1}{2} ln ∣ x + 2 ∣

= 8 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣)

Add a constant to the solution

= 8 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x + 2 ∣) + C

x \to 0 lim (\frac{sin ^{2} ( 5 x )}{x})

e^{- 9 x}, 0

t^{2} (8 e^{2 t} - 6 sin (2 t))

\frac{d}{d x} (x^{2} - 39.5 x + 120 + \frac{125}{x})

x \to 0 - lim (\frac{1}{x ^{3} - x})

What is the integral of 8/(x(x+2)) ?
The integral of 8/(x(x+2)) is 8(1/2 ln|x|-1/2 ln|x+2|)+C