What is the integral of cot(x/9) ?

The integral of cot(x/9) is 9ln|sin(x/9)|+C

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integral of cot(x/9)

\int cot (\frac{x}{9}) d x

9 ln sin (\frac{x}{9}) + C

Solution steps

\int cot (\frac{x}{9}) d x

Rewrite using trig identities

= \int \frac{cos ( \frac{x}{9} )}{sin ( \frac{x}{9} )} d x

Apply u-substitution

= \int \frac{9}{u} d u

Take the constant out:

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

= 9 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 9 ln ∣ u ∣

Substitute back

u = sin (\frac{x}{9})

= 9 ln sin (\frac{x}{9})

Add a constant to the solution

= 9 ln sin (\frac{x}{9}) + C