What is the integral of cot(x/3) ?

The integral of cot(x/3) is 3ln|sin(x/3)|+C

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integral of cot(x/3)

\int cot (\frac{x}{3}) d x

3 ln sin (\frac{x}{3}) + C

Solution steps

\int cot (\frac{x}{3}) d x

Rewrite using trig identities

= \int \frac{cos ( \frac{x}{3} )}{sin ( \frac{x}{3} )} d x

Apply u-substitution

= \int \frac{3}{u} d u

Take the constant out:

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

= 3 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 ln ∣ u ∣

Substitute back

u = sin (\frac{x}{3})

= 3 ln sin (\frac{x}{3})

Add a constant to the solution

= 3 ln sin (\frac{x}{3}) + C