What is the integral of cot(ax) ?

The integral of cot(ax) is 1/a ln|sin(ax)|+C

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integral of cot(ax)

\int cot (a x) d x

\frac{1}{a} ln ∣ sin (a x) ∣ + C

Solution steps

\int cot (a x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int \frac{1}{a u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{a} ln ∣ u ∣

Substitute back

u = sin (a x)

= \frac{1}{a} ln ∣ sin (a x) ∣

Add a constant to the solution

= \frac{1}{a} ln ∣ sin (a x) ∣ + C

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