What is the integral of cot(θ/7) ?

The integral of cot(θ/7) is 7ln|sin(θ/7)|+C

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integral of cot(θ/7)

\int cot (\frac{θ}{7}) d θ

7 ln sin (\frac{θ}{7}) + C

Solution steps

\int cot (\frac{θ}{7}) d θ

Rewrite using trig identities

= \int \frac{cos ( \frac{θ}{7} )}{sin ( \frac{θ}{7} )} d θ

Apply u-substitution

: \int \frac{7}{u} d u

= \int \frac{7}{u} d u

Take the constant out:

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

= 7 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 7 ln ∣ u ∣

Substitute back

u = sin (\frac{θ}{7})

= 7 ln sin (\frac{θ}{7})

Add a constant to the solution

= 7 ln sin (\frac{θ}{7}) + C

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