What is the integral of cot(θ/5) ?

The integral of cot(θ/5) is 5ln|sin(θ/5)|+C

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integral of cot(θ/5)

\int cot (\frac{θ}{5}) d θ

5 ln sin (\frac{θ}{5}) + C

Solution steps

\int cot (\frac{θ}{5}) d θ

Rewrite using trig identities

= \int \frac{cos ( \frac{θ}{5} )}{sin ( \frac{θ}{5} )} d θ

Apply u-substitution

: \int \frac{5}{u} d u

= \int \frac{5}{u} d u

Take the constant out:

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

= 5 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 ln ∣ u ∣

Substitute back

u = sin (\frac{θ}{5})

= 5 ln sin (\frac{θ}{5})

Add a constant to the solution

= 5 ln sin (\frac{θ}{5}) + C

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