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

The integral of cot(x/a) is aln|sin(x/a)|+C

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integral of cot(x/a)

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

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

Solution steps

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

Apply u-substitution

= \int cot (u) a d u

Take the constant out:

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

= a \cdot \int cot (u) d u

Rewrite using trig identities

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= a ln ∣ v ∣

Substitute back

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

Add a constant to the solution

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