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

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

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

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

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

Solution steps

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

Rewrite using trig identities

= \int \frac{e ^{- x}}{sin ( e ^{- x} )} d x

Apply u-substitution

= \int - csc (u) d u

Take the constant out:

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

= - \int csc (u) d u

Apply u-substitution

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - ln ∣ v ∣

Substitute back

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

Add a constant to the solution

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

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