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

The integral of cot(x/(a-b)) is (a-b)ln|sin(x/(a-b))|+C

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integral of cot(x/(a-b))

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

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

Solution steps

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

Apply u-substitution

: \int cot (u) (a - b) d u

= \int cot (u) (a - b) d u

Take the constant out:

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

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

\int cot (u) d u = ln ∣ sin (u) ∣

= (a - b) ln ∣ sin (u) ∣

Substitute back

u = \frac{x}{a - b}

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

Add a constant to the solution

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

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What is the integral of cot(x/(a-b)) ?
The integral of cot(x/(a-b)) is (a-b)ln|sin(x/(a-b))|+C