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

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

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integral of 1/(1-2u)

\int \frac{1}{1 - 2 u} d u

- \frac{1}{2} ln ∣ 1 - 2 u ∣ + C

Solution steps

\int \frac{1}{1 - 2 u} d u

Apply u-substitution

: \int - \frac{1}{2 v} d v

= \int - \frac{1}{2 v} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - \frac{1}{2} ln ∣ v ∣

Substitute back

v = 1 - 2 u

= - \frac{1}{2} ln ∣ 1 - 2 u ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ 1 - 2 u ∣ + C

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