integral from 1 to t of xe^{-x^2}

Lösung

\int_{1}^{t} x e^{- x^{2}} d x

- \frac{1}{2} (e^{- t^{2}} - \frac{1}{e})

Schritte zur Lösung

\int_{1}^{t} x e^{- x^{2}} d x

Wende U-Substitution an

= \int_{- 1}^{- t^{2}} - \frac{e ^{u}}{2} d u

Entferne die Konstante:

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

= - \frac{1}{2} \cdot \int_{- 1}^{- t^{2}} e^{u} d u

Nutze das gemeinsame Integral :

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

= - \frac{1}{2} [e^{u}]_{- 1}^{- t^{2}}

Berechne die Grenzen:

e^{- t^{2}} - \frac{1}{e}

= - \frac{1}{2} (e^{- t^{2}} - \frac{1}{e})

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