What is the integral of csc(3t) ?

The integral of csc(3t) is 1/3 ln|tan((3t)/2)|+C

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integral of csc(3t)

\int csc (3 t) d t

\frac{1}{3} ln tan (\frac{3 t}{2}) + C

Solution steps

\int csc (3 t) d t

Apply u-substitution

: \int csc (u) \frac{1}{3} d u

= \int csc (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int csc (u) d u

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{3} ln ∣ v ∣

Substitute back

= \frac{1}{3} ln tan (\frac{3 t}{2})

Add a constant to the solution

= \frac{1}{3} ln tan (\frac{3 t}{2}) + C

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What is the integral of csc(3t) ?
The integral of csc(3t) is 1/3 ln|tan((3t)/2)|+C