What is the integral of xcsc(3x^2)cot(3x^2) ?

The integral of xcsc(3x^2)cot(3x^2) is -1/6 csc(3x^2)+C

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integral of xcsc(3x^2)cot(3x^2)

\int x csc (3 x^{2}) cot (3 x^{2}) d x

- \frac{1}{6} csc (3 x^{2}) + C

Solution steps

\int x csc (3 x^{2}) cot (3 x^{2}) d x

Apply u-substitution

: \int \frac{csc ( u ) cot ( u )}{6} d u

= \int \frac{csc ( u ) cot ( u )}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int csc (u) cot (u) d u

Apply u-substitution

: \int - 1 d v

= \frac{1}{6} \cdot \int - 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{6} (- 1) v

Substitute back

= \frac{1}{6} (- 1) csc (3 x^{2})

Simplify

= - \frac{1}{6} csc (3 x^{2})

Add a constant to the solution

= - \frac{1}{6} csc (3 x^{2}) + C

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What is the integral of xcsc(3x^2)cot(3x^2) ?
The integral of xcsc(3x^2)cot(3x^2) is -1/6 csc(3x^2)+C