What is the integral of 3xcsc(3x)cot(3x) ?

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

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integral of 3xcsc(3x)cot(3x)

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

- x csc (3 x) + \frac{1}{3} ln tan (\frac{3 x}{2}) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{1}{3} csc (3 x) d x = - \frac{1}{9} ln tan (\frac{3 x}{2})

= 3 (- \frac{1}{3} x csc (3 x) - (- \frac{1}{9} ln tan (\frac{3 x}{2})))

Simplify

3 (- \frac{1}{3} x csc (3 x) - (- \frac{1}{9} ln tan (\frac{3 x}{2}))) : - x csc (3 x) + \frac{1}{3} ln tan (\frac{3 x}{2})

= - x csc (3 x) + \frac{1}{3} ln tan (\frac{3 x}{2})

Add a constant to the solution

= - x csc (3 x) + \frac{1}{3} ln tan (\frac{3 x}{2}) + C

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