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

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

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integral of (1-cos(x))/(x-sin(x))

\int \frac{1 - cos ( x )}{x - sin ( x )} d x

ln ∣ x - sin (x) ∣ + C

Solution steps

\int \frac{1 - cos ( x )}{x - sin ( x )} d x

Apply u-substitution

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

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = x - sin (x)

= ln ∣ x - sin (x) ∣

Add a constant to the solution

= ln ∣ x - sin (x) ∣ + C

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