integral from 1 to 5 of e^{-x}

Lösung

\int_{1}^{5} e^{- x} d x

\frac{1}{e} - \frac{1}{e ^{5}}

Dezimale

0.36114 \dots

Schritte zur Lösung

\int_{1}^{5} e^{- x} d x

Wende U-Substitution an

= \int_{- 1}^{- 5} - e^{u} d u

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

= - \int_{- 5}^{- 1} - e^{u} d u

Entferne die Konstante:

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

= - (- \int_{- 5}^{- 1} e^{u} d u)

Nutze das gemeinsame Integral :

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

= - (- [e^{u}]_{- 5}^{- 1})

Vereinfache

= [e^{u}]_{- 5}^{- 1}

Berechne die Grenzen:

\frac{1}{e} - \frac{1}{e ^{5}}

= \frac{1}{e} - \frac{1}{e ^{5}}

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