What is the integral of e^θcsc(e^θ+1) ?

The integral of e^θcsc(e^θ+1) is ln|tan((e^θ+1)/2)|+C

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integral of e^θcsc(e^θ+1)

\int e^{θ} csc (e^{θ} + 1) d θ

ln tan (\frac{e ^{θ} + 1}{2}) + C

Solution steps

\int e^{θ} csc (e^{θ} + 1) d θ

Apply u-substitution

= \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 ^{θ} + 1}{2})

Add a constant to the solution

= ln tan (\frac{e ^{θ} + 1}{2}) + C

What is the integral of e^θcsc(e^θ+1) ?
The integral of e^θcsc(e^θ+1) is ln|tan((e^θ+1)/2)|+C