What is the integral of cot(x/2) ?

The integral of cot(x/2) is 2ln|sin(x/2)|+C

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integral of cot(x/2)

integral

cot (\frac{x}{2})

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

Solution steps

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

Rewrite using trig identities

= \int \frac{cos ( \frac{x}{2} )}{sin ( \frac{x}{2} )} d x

Apply u-substitution

= \int \frac{2}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = sin (\frac{x}{2})

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

Add a constant to the solution

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