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

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

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

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

ln ∣ sin (x) - x cos (x) ∣ + C

Solution steps

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

Rewrite using trig identities

= \int \frac{x sin ( x )}{sin ( x ) - x cos ( 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 = sin (x) - x cos (x)

= ln ∣ sin (x) - x cos (x) ∣

Add a constant to the solution

= ln ∣ sin (x) - x cos (x) ∣ + C

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