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

The integral of e^{2x}csc(e^{2x}) is 1/2 ln|tan((e^{2x})/2)|+C

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

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

\frac{1}{2} ln tan (\frac{e ^{2 x}}{2}) + C

Solution steps

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

Apply u-substitution

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

= \int \frac{csc ( 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) d u

Apply u-substitution

: \int \frac{1}{v} d v

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

= \frac{1}{2} ln tan (\frac{e ^{2 x}}{2})

Add a constant to the solution

= \frac{1}{2} ln tan (\frac{e ^{2 x}}{2}) + C

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