What is the integral of (e^{x/2})/(sqrt(1-e^x)) ?

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

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integral of (e^{x/2})/(sqrt(1-e^x))

\int \frac{e ^{\frac{x}{2}}}{1 - e ^{x}} d x

2 arcsin (e^{\frac{x}{2}}) + C

Solution steps

\int \frac{e ^{\frac{x}{2}}}{1 - e ^{x}} d x

Apply u-substitution

: \int \frac{2 e ^{u}}{1 - e ^{2 u}} d u

= \int \frac{2 e ^{u}}{1 - e ^{2 u}} d u

Take the constant out:

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

= 2 \cdot \int \frac{e ^{u}}{1 - e ^{2 u}} d u

Apply u-substitution

: \int \frac{1}{1 - v ^{2}} d v

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

Use the common integral:

\int \frac{1}{1 - v ^{2}} d v = arcsin (v)

= 2 arcsin (v)

Substitute back

= 2 arcsin (e^{\frac{x}{2}})

Add a constant to the solution

= 2 arcsin (e^{\frac{x}{2}}) + C

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