What is the integral of cot((4z)/3) ?

The integral of cot((4z)/3) is 3/4 ln|sin((4z)/3)|+C

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integral of cot((4z)/3)

\int cot (\frac{4 z}{3}) d z

\frac{3}{4} ln sin (\frac{4 z}{3}) + C

Solution steps

\int cot (\frac{4 z}{3}) d z

Apply u-substitution

: \int cot (u) \frac{3}{4} d u

= \int cot (u) \frac{3}{4} d u

Take the constant out:

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

= \frac{3}{4} \cdot \int cot (u) d u

Rewrite using trig identities

= \frac{3}{4} \cdot \int \frac{cos ( u )}{sin ( u )} d u

Apply u-substitution

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

= \frac{3}{4} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{3}{4} ln ∣ v ∣

Substitute back

= \frac{3}{4} ln sin (\frac{4 z}{3})

Add a constant to the solution

= \frac{3}{4} ln sin (\frac{4 z}{3}) + C

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What is the integral of cot((4z)/3) ?
The integral of cot((4z)/3) is 3/4 ln|sin((4z)/3)|+C