What is the integral of e^{3x}csc(e^{3x})cot(e^{3x}) ?

The integral of e^{3x}csc(e^{3x})cot(e^{3x}) is -1/3 csc(e^{3x})+C

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integral of e^{3x}csc(e^{3x})cot(e^{3x})

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int - 1 d v

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

Integral of a constant:

\int a d x = a x

= \frac{1}{3} (- 1) v

Substitute back

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

Simplify

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

Add a constant to the solution

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

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What is the integral of e^{3x}csc(e^{3x})cot(e^{3x}) ?
The integral of e^{3x}csc(e^{3x})cot(e^{3x}) is -1/3 csc(e^{3x})+C