What is the integral of (1-2x)/(x(1-x)) ?

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

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integral of (1-2x)/(x(1-x))

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

ln ∣ x ∣ - ln ∣ x - 1 ∣ + 2 ln ∣ 1 - x ∣ + C

Solution steps

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

Expand

\frac{1 - 2 x}{x ( 1 - x )} : \frac{1}{x ( 1 - x )} - \frac{2 x}{x ( 1 - 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 ( 1 - x )} d x - \int \frac{2 x}{x ( 1 - x )} d x

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

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

= ln ∣ x ∣ - ln ∣ x - 1 ∣ - (- 2 ln ∣ 1 - x ∣)

Simplify

= ln ∣ x ∣ - ln ∣ x - 1 ∣ + 2 ln ∣ 1 - x ∣

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ x - 1 ∣ + 2 ln ∣ 1 - x ∣ + C

\frac{- 4 x}{x ^{2} + 1}

\frac{d}{d x} ((x - 4)^{2})

\int \frac{( 1 - x ) ^{3}}{x ^{3} x} d x

\frac{d}{d t} (e^{3 t} cos (2 t))

f (x) = ln (4 - x^{8})

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