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Beliebt Rechnen >

integral of x^5e^{4x}

Lösung

\int x^{5} e^{4 x} d x

Lösung

\frac{1}{4096} (1024 e^{4 x} x^{5} - 5 (256 e^{4 x} x^{4} - 4 (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x}))))) + C

Schritte zur Lösung

\int x^{5} e^{4 x} d x

Wende U-Substitution an

= \int \frac{e ^{u} u ^{5}}{4096} d u

Entferne die Konstante:

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

= \frac{1}{4096} \cdot \int e^{u} u^{5} d u

Wende die partielle Integration an

= \frac{1}{4096} (u^{5} e^{u} - \int 5 u^{4} e^{u} d u)

\int 5 u^{4} e^{u} d u = 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))))

= \frac{1}{4096} (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))))

Setze in

u = 4 x

ein

= \frac{1}{4096} ((4 x)^{5} e^{4 x} - 5 ((4 x)^{4} e^{4 x} - 4 ((4 x)^{3} e^{4 x} - 3 ((4 x)^{2} e^{4 x} - 2 (e^{4 x} \cdot 4 x - e^{4 x})))))

Vereinfache

\frac{1}{4096} ((4 x)^{5} e^{4 x} - 5 ((4 x)^{4} e^{4 x} - 4 ((4 x)^{3} e^{4 x} - 3 ((4 x)^{2} e^{4 x} - 2 (e^{4 x} \cdot 4 x - e^{4 x}))))) : \frac{1}{4096} (1024 e^{4 x} x^{5} - 5 (256 e^{4 x} x^{4} - 4 (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x})))))

= \frac{1}{4096} (1024 e^{4 x} x^{5} - 5 (256 e^{4 x} x^{4} - 4 (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x})))))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4096} (1024 e^{4 x} x^{5} - 5 (256 e^{4 x} x^{4} - 4 (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x}))))) + C

Graph

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