What is the integral of cot(2u) ?

The integral of cot(2u) is 1/2 ln|sin(2u)|+C

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integral of cot(2u)

\int cot (2 u) d u

\frac{1}{2} ln ∣ sin (2 u) ∣ + C

Solution steps

\int cot (2 u) d u

Rewrite using trig identities

= \int \frac{cos ( 2 u )}{sin ( 2 u )} d u

Apply u-substitution

: \int \frac{1}{2 v} d v

= \int \frac{1}{2 v} d v

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

v = sin (2 u)

= \frac{1}{2} ln ∣ sin (2 u) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ sin (2 u) ∣ + C

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