What is the integral of 1/(3x^2-3x) ?

The integral of 1/(3x^2-3x) is 1/(3(x-1))+(-2-(ln|x|-ln|x-1|-1/(x-1))(x-1))/(3(x-1))+C

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integral of 1/(3x^2-3x)

integral

\frac{1}{3 x ^{2} - 3 x}

\frac{1}{3 ( x - 1 )} + \frac{- 2 - ( ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1} ) ( x - 1 )}{3 ( x - 1 )} + C

Solution steps

\int \frac{1}{3 x ^{2} - 3 x} d x

Apply Integration By Parts

= \frac{1}{3 ( x - 1 )} - \int - \frac{2 x - 1}{3 x ( x - 1 ) ^{2}} d x

\int - \frac{2 x - 1}{3 x ( x - 1 ) ^{2}} d x = - \frac{- 2 - ( ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1} ) ( x - 1 )}{3 ( x - 1 )}

= \frac{1}{3 ( x - 1 )} - (- \frac{- 2 - ( ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1} ) ( x - 1 )}{3 ( x - 1 )})

Simplify

= \frac{1}{3 ( x - 1 )} + \frac{- 2 - ( ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1} ) ( x - 1 )}{3 ( x - 1 )}

Add a constant to the solution

= \frac{1}{3 ( x - 1 )} + \frac{- 2 - ( ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1} ) ( x - 1 )}{3 ( x - 1 )} + C

What is the integral of 1/(3x^2-3x) ?
The integral of 1/(3x^2-3x) is 1/(3(x-1))+(-2-(ln|x|-ln|x-1|-1/(x-1))(x-1))/(3(x-1))+C