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

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

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

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

ln ∣ x ∣ - ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

= \int \frac{1}{x} - \frac{1}{x - 1} 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}{x} d x - \int \frac{1}{x - 1} d x

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - ln ∣ x - 1 ∣

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ x - 1 ∣ + C

f (x) = x + 5

sin (2 x), π

\int_{0}^{π} x \cdot sin (2 x) d x

\frac{d y}{d x} = (x - 2) e^{- 2 y}, y (2) = ln (2)

\int_{0}^{\frac{π}{3}} sin (3 x) d x

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