integral from 0 to 1 of e^{x-e^x}

Lösung

\int_{0}^{1} e^{x - e^{x}} d x

\frac{1}{e} - \frac{1}{e ^{e}}

Dezimale

0.30189 \dots

Schritte zur Lösung

\int_{0}^{1} e^{x - e^{x}} d x

Wende U-Substitution an

= \int_{1}^{e} e^{- u} d u

Wende U-Substitution an

= \int_{- 1}^{- e} - e^{v} d v

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- e}^{- 1} - e^{v} d v

Entferne die Konstante:

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

= - (- \int_{- e}^{- 1} e^{v} d v)

Nutze das gemeinsame Integral :

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

= - (- [e^{v}]_{- e}^{- 1})

Vereinfache

= [e^{v}]_{- e}^{- 1}

Berechne die Grenzen:

\frac{1}{e} - \frac{1}{e ^{e}}

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

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