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Beliebt Rechnen >

integral of (3x^3+4x)e^{x^2}

Lösung

\int (3 x^{3} + 4 x) e^{x^{2}} d x

Lösung

\frac{3}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

Schritte zur Lösung

\int (3 x^{3} + 4 x) e^{x^{2}} d x

Multipliziere aus

(3 x^{3} + 4 x) e^{x^{2}} : 3 e^{x^{2}} x^{3} + 4 e^{x^{2}} x

= \int 3 e^{x^{2}} x^{3} + 4 e^{x^{2}} x d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 3 e^{x^{2}} x^{3} d x + \int 4 e^{x^{2}} x d x

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

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

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

Vereinfache

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

= \frac{3}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

Graph

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