What is the integral of e^{2x}csc(e^{2x})cot(e^{2x}) ?

The integral of e^{2x}csc(e^{2x})cot(e^{2x}) is -1/2 csc(e^{2x})+C

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integral of e^{2x}csc(e^{2x})cot(e^{2x})

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

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

Solution steps

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

Apply u-substitution

: \int \frac{csc ( u ) cot ( u )}{2} d u

= \int \frac{csc ( u ) cot ( u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int csc (u) cot (u) d u

Apply u-substitution

: \int - 1 d v

= \frac{1}{2} \cdot \int - 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{2} (- 1) v

Substitute back

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

Simplify

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

Add a constant to the solution

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

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What is the integral of e^{2x}csc(e^{2x})cot(e^{2x}) ?
The integral of e^{2x}csc(e^{2x})cot(e^{2x}) is -1/2 csc(e^{2x})+C