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

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

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

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

- x cot (\frac{x}{2}) + 2 ln sin (\frac{x}{2}) + C

Solution steps

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

Apply Integration By Parts

: - x cot (\frac{x}{2}) - \int - cot (\frac{x}{2}) d x

= - x cot (\frac{x}{2}) - \int - cot (\frac{x}{2}) d x

\int - cot (\frac{x}{2}) d x = - 2 ln sin (\frac{x}{2})

= - x cot (\frac{x}{2}) - (- 2 ln sin (\frac{x}{2}))

Simplify

= - x cot (\frac{x}{2}) + 2 ln sin (\frac{x}{2})

Add a constant to the solution

= - x cot (\frac{x}{2}) + 2 ln sin (\frac{x}{2}) + C

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

\int \frac{x ^{3} - 4}{x ^{3}} d x

\int 2 x^{3} e^{x^{4}} d x

cot^{2} (x)

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