What is the integral of 1/(y(1-y)) ?

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

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integral of 1/(y(1-y))

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

ln ∣ y ∣ - ln ∣ y - 1 ∣ + C

Solution steps

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

Take the partial fraction of

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

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

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}{y} d y - \int \frac{1}{y - 1} d y

\int \frac{1}{y} d y = ln ∣ y ∣

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

= ln ∣ y ∣ - ln ∣ y - 1 ∣

Add a constant to the solution

= ln ∣ y ∣ - ln ∣ y - 1 ∣ + C

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