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Beliebt Rechnen >

integral from 0 to 2 of pi(e^{-x^2})^2

Lösung

\int_{0}^{2} π (e^{- x^{2}})^{2} d x

Lösung

1.96857 \dots

Schritte zur Lösung

\int_{0}^{2} π (e^{- x^{2}})^{2} d x

Entferne die Konstante:

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

= π \cdot \int_{0}^{2} (e^{- x^{2}})^{2} d x

Wende die partielle Integration an

= π [e^{- 2 x^{2}} x - \int - 4 e^{- 2 x^{2}} x^{2} d x]_{0}^{2}

\int - 4 e^{- 2 x^{2}} x^{2} d x = e^{- 2 x^{2}} x - \frac{π}{2 2} erf (2 x)

= π [e^{- 2 x^{2}} x - (e^{- 2 x^{2}} x - \frac{π}{2 2} erf (2 x))]_{0}^{2}

Vereinfache

= π [\frac{π}{2 2} erf (2 x)]_{0}^{2}

Berechne die Grenzen:

0.62661 \dots

= π 0.62661 \dots

Vereinfache

= 1.96857 \dots

Graph

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