What is the integral of cot(3x)+csc(3x) ?

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

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integral of cot(3x)+csc(3x)

\int cot (3 x) + csc (3 x) d x

\frac{1}{3} ln ∣ sin (3 x) ∣ + \frac{1}{3} ln tan (\frac{3 x}{2}) + C

Solution steps

\int cot (3 x) + csc (3 x) d x

Rewrite using trig identities

= \int \frac{1}{tan ( 3 x )} + csc (3 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{tan ( 3 x )} d x + \int csc (3 x) d x

\int \frac{1}{tan ( 3 x )} d x = \frac{1}{3} ln ∣ sin (3 x) ∣

\int csc (3 x) d x = \frac{1}{3} ln tan (\frac{3 x}{2})

= \frac{1}{3} ln ∣ sin (3 x) ∣ + \frac{1}{3} ln tan (\frac{3 x}{2})

Add a constant to the solution

= \frac{1}{3} ln ∣ sin (3 x) ∣ + \frac{1}{3} ln tan (\frac{3 x}{2}) + C

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