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

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

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integral of ln(1-2x)

\int ln (1 - 2 x) d x

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

Solution steps

\int ln (1 - 2 x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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x \to a lim (\frac{1}{x})

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