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

The integral of e^xcsc^2(e^x) is -cot(e^x)+C

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integral of e^xcsc^2(e^x)

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

- cot (e^{x}) + C

Solution steps

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

Apply u-substitution

: \int csc^{2} (u) d u

= \int csc^{2} (u) d u

Use the common integral:

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

= - cot (u)

Substitute back

u = e^{x}

= - cot (e^{x})

Add a constant to the solution

= - cot (e^{x}) + C

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What is the integral of e^xcsc^2(e^x) ?
The integral of e^xcsc^2(e^x) is -cot(e^x)+C