integral from 0 to 5 of 0.5e^{-0.5x}

Lösung

\int_{0}^{5} 0.5 e^{- 0.5 x} d x

0.91791 \dots

Schritte zur Lösung

\int_{0}^{5} 0.5 e^{- 0.5 x} d x

Entferne die Konstante:

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

= 0.5 \cdot \int_{0}^{5} e^{- 0.5 x} d x

Wende U-Substitution an

= 0.5 \cdot \int_{0}^{- 2.5} - \frac{1}{0.5} e^{u} d u

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

= 0.5 (- \int_{- 2.5}^{0} - \frac{1}{0.5} e^{u} d u)

Entferne die Konstante:

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

= 0.5 (- (- \frac{1}{0.5} \cdot \int_{- 2.5}^{0} e^{u} d u))

Vereinfache

= 0.5 (- (- 2 \cdot \int_{- 2.5}^{0} e^{u} d u))

Nutze das gemeinsame Integral :

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

= 0.5 (- (- 2 [e^{u}]_{- 2.5}^{0}))

Vereinfache

= [e^{u}]_{- 2.5}^{0}

Berechne die Grenzen:

0.91791 \dots

= 0.91791 \dots

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