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

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

\int x^{3} x^{2} + 21 d x

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

\int \frac{2}{x ^{2} ( x ^{2} - 4 x - 5 )} d x

\int x^{3} sin^{2} (4 x) d x

\int \frac{( 4 x + 3 )}{4 x ^{3} + 8 x ^{2} + 3 x} d x

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