What is the integral of cot(39x) ?

The integral of cot(39x) is 1/39 ln|sin(39x)|+C

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integral of cot(39x)

\int cot (39 x) d x

\frac{1}{39} ln ∣ sin (39 x) ∣ + C

Solution steps

\int cot (39 x) d x

Rewrite using trig identities

= \int \frac{cos ( 39 x )}{sin ( 39 x )} d x

Apply u-substitution

= \int \frac{1}{39 u} d u

Take the constant out:

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

= \frac{1}{39} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{39} ln ∣ u ∣

Substitute back

u = sin (39 x)

= \frac{1}{39} ln ∣ sin (39 x) ∣

Add a constant to the solution

= \frac{1}{39} ln ∣ sin (39 x) ∣ + C

What is the integral of cot(39x) ?
The integral of cot(39x) is 1/39 ln|sin(39x)|+C