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

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

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

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

- \frac{1}{2} ln ∣ 1 - x ∣ + C

Solution steps

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

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}{1 - x} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} (- ln ∣ u ∣)

Substitute back

u = 1 - x

= \frac{1}{2} (- ln ∣ 1 - x ∣)

Simplify

= - \frac{1}{2} ln ∣ 1 - x ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ 1 - x ∣ + C

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