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

The integral of 3/(sin(3x)) is ln|tan((3x)/2)|+C

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integral of 3/(sin(3x))

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

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

Solution steps

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

Take the constant out:

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

= 3 \cdot \int \frac{1}{sin ( 3 x )} d x

Rewrite using trig identities

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

Apply u-substitution

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

= 3 \cdot \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

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 \cdot \frac{1}{3} ln ∣ v ∣

Substitute back

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

Simplify

3 \cdot \frac{1}{3} ln tan (\frac{3 x}{2}) : ln tan (\frac{3 x}{2})

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

Add a constant to the solution

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

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What is the integral of 3/(sin(3x)) ?
The integral of 3/(sin(3x)) is ln|tan((3x)/2)|+C