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Beliebt Rechnen >

integral of x^2e^{-(x^2)/2}

Lösung

\int x^{2} e^{- \frac{x ^{2}}{2}} d x

Lösung

- e^{- \frac{x ^{2}}{2}} x + (\frac{π}{2})^{\frac{1}{2}} erf (\frac{x}{2}) + C

Schritte zur Lösung

\int x^{2} e^{- \frac{x ^{2}}{2}} d x

Vereinfache

x^{2} : xx

= \int xx e^{- \frac{x ^{2}}{2}} d x

Wende die partielle Integration an

= - e^{- \frac{x ^{2}}{2}} x - \int - e^{- \frac{x ^{2}}{2}} d x

\int - e^{- \frac{x ^{2}}{2}} d x = - (\frac{π}{2})^{\frac{1}{2}} erf (\frac{x}{2})

= - e^{- \frac{x ^{2}}{2}} x - (- (\frac{π}{2})^{\frac{1}{2}} erf (\frac{x}{2}))

Vereinfache

= - e^{- \frac{x ^{2}}{2}} x + (\frac{π}{2})^{\frac{1}{2}} erf (\frac{x}{2})

Füge eine Konstante zur Lösung hinzu

= - e^{- \frac{x ^{2}}{2}} x + (\frac{π}{2})^{\frac{1}{2}} erf (\frac{x}{2}) + C

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