What is the integral of csc(x/3) ?

The integral of csc(x/3) is 3ln|tan(x/6)|+C

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integral of csc(x/3)

\int csc (\frac{x}{3}) d x

3 ln tan (\frac{x}{6}) + C

Solution steps

\int csc (\frac{x}{3}) d x

Apply u-substitution

: \int csc (u) \cdot 3 d u

= \int csc (u) \cdot 3 d u

Take the constant out:

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

= 3 \cdot \int csc (u) d u

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 ln ∣ v ∣

Substitute back

= 3 ln tan (\frac{\frac{x}{3}}{2})

\frac{\frac{x}{3}}{2} = \frac{x}{6}

= 3 ln tan (\frac{x}{6})

Add a constant to the solution

= 3 ln tan (\frac{x}{6}) + C

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