Integrate [x - sin x] / [1 - cos x].

The method of calculating the antiderivative of a function is called integration.

Answer: After integrating [x - sin x] / [1 - cos x], we get -x cot (x/2) + C

With the help of trigonometric formulas, we will find the solution.

Explanation:

We have, ∫ [x - sin x] / [1 - cos x] dx

Applying double angle formulas, 

= ∫  (x - 2 sin (x/2) cos (x/2)) / (2 sin²(x/2))  dx

= ∫  (x / (2 sin²(x/2)) - 2 (sin (x/2) cos (x/2)) / (2 sin²(x/2))  dx

= ∫ (1/2) x cosec²(x/2) - cot (x/2) dx  ……(1)

= 1/2 ∫ [-x (2cot (x/2)) - ∫ {(d/dx) x ∫ cosec²(x/2) dx} dx] - ∫ cot (x/2) dx

= -x cot (x/2) + ∫ cot (x/2) dx - ∫ cot (x/2) dx + C

= -x cot (x/2) + C

Thus, the integration of [x - sin x] / [1 - cos x] is equal to -x cot (x/2) + C