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Beliebt Rechnen >

integral from 0 to N of e^{-st}e^{-3t}

Lösung

\int_{0}^{N} e^{- s t} e^{- 3 t} d t

Lösung

\frac{e ^{- N (s + 3)} - 1}{- s - 3}

Schritte zur Lösung

\int_{0}^{N} e^{- s t} e^{- 3 t} d t

Faktorisiere

e^{- 3 t} e^{- s t} : e^{- t (s + 3)}

= \int_{0}^{N} e^{- t (s + 3)} d t

Wende U-Substitution an

= \int_{0}^{- N (s + 3)} \frac{e ^{u}}{- s - 3} d u

Entferne die Konstante:

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

= \frac{1}{- s - 3} \cdot \int_{0}^{- N (s + 3)} e^{u} d u

Nutze das gemeinsame Integral :

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

= \frac{1}{- s - 3} [e^{u}]_{0}^{- N (s + 3)}

Vereinfache

\frac{1}{- s - 3} [e^{u}]_{0}^{- N (s + 3)} : \frac{[ e ^{u} ] _{0}^{- N (s + 3)}}{- s - 3}

= \frac{[ e ^{u} ] _{0}^{- N (s + 3)}}{- s - 3}

Berechne die Grenzen:

e^{- N (s + 3)} - 1

= \frac{e ^{- N (s + 3)} - 1}{- s - 3}

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