What is the integral of (1/(x(x-4))) ?

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

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integral of (1/(x(x-4)))

\int (\frac{1}{x ( x - 4 )}) d x

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

Solution steps

\int \frac{1}{x ( x - 4 )} d x

Take the partial fraction of

\frac{1}{x ( x - 4 )} : - \frac{1}{4 x} + \frac{1}{4 ( x - 4 )}

= \int - \frac{1}{4 x} + \frac{1}{4 ( 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

= - \int \frac{1}{4 x} d x + \int \frac{1}{4 ( x - 4 )} d x

\int \frac{1}{4 x} d x = \frac{1}{4} ln ∣ x ∣

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

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

Add a constant to the solution

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

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f (x) = (12 x - 3)^{\frac{1}{2}}, at x = 1

x \to - \infty lim (e^{x} - x)

\frac{d}{d x} (cot^{2} (5 x))

\frac{d}{d x} (cos (3 sin (x)))

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