What is the integral of e^{2t-e^{2t}} ?

The integral of e^{2t-e^{2t}} is -1/2 e^{-e^{2t}}+C

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integral of e^{2t-e^{2t}}

\int e^{2 t - e^{2 t}} d t

- \frac{1}{2} e^{- e^{2 t}} + C

Solution steps

\int e^{2 t - e^{2 t}} d t

Apply u-substitution

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

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

Apply u-substitution

: \int e^{- v} d v

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

Apply u-substitution

: \int - e^{w} d w

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

Take the constant out:

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

= \frac{1}{2} (- \int e^{w} d w)

Use the common integral:

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

= \frac{1}{2} (- e^{w})

Substitute back

= \frac{1}{2} (- e^{- e^{2 t}})

Simplify

= - \frac{1}{2} e^{- e^{2 t}}

Add a constant to the solution

= - \frac{1}{2} e^{- e^{2 t}} + C

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What is the integral of e^{2t-e^{2t}} ?
The integral of e^{2t-e^{2t}} is -1/2 e^{-e^{2t}}+C