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Beliebt Rechnen >

integral of x^2e^{sqrt(x)}

Lösung

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

Lösung

2 (e^{x} x^{\frac{5}{2}} - 5 (e^{x} x^{2} - 4 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x}))))) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int 2 e^{u} u^{5} d u

Entferne die Konstante:

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

= 2 \cdot \int e^{u} u^{5} d u

Wende die partielle Integration an

= 2 (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}))))

= 2 (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 = x

ein

= 2 ((x)^{5} e^{x} - 5 ((x)^{4} e^{x} - 4 ((x)^{3} e^{x} - 3 ((x)^{2} e^{x} - 2 (e^{x} x - e^{x})))))

Vereinfache

2 ((x)^{5} e^{x} - 5 ((x)^{4} e^{x} - 4 ((x)^{3} e^{x} - 3 ((x)^{2} e^{x} - 2 (e^{x} x - e^{x}))))) : 2 (e^{x} x^{\frac{5}{2}} - 5 (e^{x} x^{2} - 4 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x})))))

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

Füge eine Konstante zur Lösung hinzu

= 2 (e^{x} x^{\frac{5}{2}} - 5 (e^{x} x^{2} - 4 (e^{x} x^{\frac{3}{2}} - 3 (e^{x} x - 2 (e^{x} x - e^{x}))))) + C

Graph

Beliebte Beispiele

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\int 5 x^{4} cos (x) d x

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\int (x^{\frac{5}{9}} - 7 x^{\frac{7}{6}}) d x

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\int x^{7} sec^{2} (x^{8}) d x

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\int \frac{10 ( x - 9 )}{( x + 5 ) ( x - 2 )} d x

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\int \frac{1}{x ^{2}} + \frac{1}{x ^{3}} d x