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Beliebt Rechnen >

integral of x^4e^{-2x}

Lösung

\int x^{4} e^{- 2 x} d x

Lösung

- \frac{1}{4} e^{- 2 x} (2 x^{4} + 4 x^{3} + 3 (2 x^{2} + 2 x + 1)) + C

Schritte zur Lösung

\int x^{4} e^{- 2 x} d x

Wende U-Substitution an

= \int - \frac{e ^{u} u ^{4}}{32} d u

Entferne die Konstante:

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

= - \frac{1}{32} \cdot \int e^{u} u^{4} d u

Wende die partielle Integration an

= - \frac{1}{32} (u^{4} e^{u} - \int 4 u^{3} e^{u} d u)

\int 4 u^{3} e^{u} d u = 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

= - \frac{1}{32} (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))))

Setze in

u = - 2 x

ein

= - \frac{1}{32} ((- 2 x)^{4} e^{- 2 x} - 4 ((- 2 x)^{3} e^{- 2 x} - 3 ((- 2 x)^{2} e^{- 2 x} - 2 (e^{- 2 x} (- 2 x) - e^{- 2 x}))))

Vereinfache

- \frac{1}{32} ((- 2 x)^{4} e^{- 2 x} - 4 ((- 2 x)^{3} e^{- 2 x} - 3 ((- 2 x)^{2} e^{- 2 x} - 2 (e^{- 2 x} (- 2 x) - e^{- 2 x})))) : - \frac{1}{4} e^{- 2 x} (2 x^{4} + 4 x^{3} + 3 (2 x^{2} + 2 x + 1))

= - \frac{1}{4} e^{- 2 x} (2 x^{4} + 4 x^{3} + 3 (2 x^{2} + 2 x + 1))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} e^{- 2 x} (2 x^{4} + 4 x^{3} + 3 (2 x^{2} + 2 x + 1)) + C

Graph

Beliebte Beispiele

integral from-4 to 2 of x^2+2

\int_{- 4}^{2} x^{2} + 2 d x

derivative of (Ax+B/(Cx+d))

\frac{d}{d x} (\frac{A x + B}{C x + d})

integral from-1 to 3 of 3x^3

\int_{- 1}^{3} 3 x^{3} d x

derivative of 2x^2-x^4

\frac{d}{d x} (2 x^{2} - x^{4})

derivative of 5x^2-2x

\frac{d}{d x} (5 x^{2} - 2 x)