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Beliebt Rechnen >

integral from 0 to 10 of 1000te^{-0.5t}

Lösung

\int_{0}^{10} 1000 t e^{- 0.5 t} d t

Lösung

- 4000 (- 1 + \frac{6}{e ^{5}})

+1

Dezimale

3838.28927 \dots

Schritte zur Lösung

\int_{0}^{10} 1000 t e^{- 0.5 t} d t

Entferne die Konstante:

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

= 1000 \cdot \int_{0}^{10} t e^{- 0.5 t} d t

Wende U-Substitution an

= 1000 \cdot \int_{0}^{- 5} \frac{2}{0.5} e^{u} u d u

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

= 1000 (- \int_{- 5}^{0} \frac{2}{0.5} e^{u} u d u)

Entferne die Konstante:

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

= 1000 (- \frac{2}{0.5} \cdot \int_{- 5}^{0} e^{u} u d u)

Vereinfache

= 1000 (- 4 \cdot \int_{- 5}^{0} e^{u} u d u)

Wende die partielle Integration an

= 1000 (- 4 [e^{u} u - \int e^{u} d u]_{- 5}^{0})

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

= 1000 (- 4 [e^{u} u - e^{u}]_{- 5}^{0})

Vereinfache

= - 4000 [e^{u} u - e^{u}]_{- 5}^{0}

Berechne die Grenzen:

- 1 + \frac{6}{e ^{5}}

= - 4000 (- 1 + \frac{6}{e ^{5}})

Graph

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