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

The integral of e^{x-e^x} is -e^{-e^x}+C

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integral of e^{x-e^x}

\int e^{x - e^{x}} d x

- e^{- e^{x}} + C

Solution steps

\int e^{x - e^{x}} d x

Apply u-substitution

: \int e^{- u} d u

= \int e^{- u} d u

Apply u-substitution

: \int - e^{v} d v

= \int - e^{v} d v

Take the constant out:

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

= - \int e^{v} d v

Use the common integral:

\int e^{v} d v = e^{v}

= - e^{v}

Substitute back

= - e^{- e^{x}}

Add a constant to the solution

= - e^{- e^{x}} + C

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