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Beliebt Rechnen >

integral of e^{12x}*x^3

Lösung

\int e^{12 x} \cdot x^{3} d x

Lösung

\frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1) + C

Schritte zur Lösung

\int e^{12 x} x^{3} d x

Wende U-Substitution an

= \int \frac{e ^{u} u ^{3}}{20736} d u

Entferne die Konstante:

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

= \frac{1}{20736} \cdot \int e^{u} u^{3} d u

Wende die partielle Integration an

= \frac{1}{20736} (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

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

= \frac{1}{20736} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Setze in

u = 12 x

ein

= \frac{1}{20736} ((12 x)^{3} e^{12 x} - 3 ((12 x)^{2} e^{12 x} - 2 (e^{12 x} \cdot 12 x - e^{12 x})))

Vereinfache

\frac{1}{20736} ((12 x)^{3} e^{12 x} - 3 ((12 x)^{2} e^{12 x} - 2 (e^{12 x} \cdot 12 x - e^{12 x}))) : \frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1)

= \frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1) + C

Graph

Beliebte Beispiele

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