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Beliebt Rechnen >

integral of x^2e^{-10x}

Lösung

\int x^{2} e^{- 10 x} d x

Lösung

- \frac{1}{500} e^{- 10 x} (50 x^{2} + 10 x + 1) + C

Schritte zur Lösung

\int x^{2} e^{- 10 x} d x

Wende U-Substitution an

= \int - \frac{e ^{u} u ^{2}}{1000} d u

Entferne die Konstante:

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

= - \frac{1}{1000} \cdot \int e^{u} u^{2} d u

Wende die partielle Integration an

= - \frac{1}{1000} (u^{2} e^{u} - \int 2 u e^{u} d u)

\int 2 u e^{u} d u = 2 (e^{u} u - e^{u})

= - \frac{1}{1000} (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Setze in

u = - 10 x

ein

= - \frac{1}{1000} ((- 10 x)^{2} e^{- 10 x} - 2 (e^{- 10 x} (- 10 x) - e^{- 10 x}))

Vereinfache

- \frac{1}{1000} ((- 10 x)^{2} e^{- 10 x} - 2 (e^{- 10 x} (- 10 x) - e^{- 10 x})) : - \frac{1}{500} e^{- 10 x} (50 x^{2} + 10 x + 1)

= - \frac{1}{500} e^{- 10 x} (50 x^{2} + 10 x + 1)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{500} e^{- 10 x} (50 x^{2} + 10 x + 1) + C

Graph

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