integral of (5x^3+6x)e^{x^2}

Lösung

\int (5 x^{3} + 6 x) e^{x^{2}} d x

\frac{5}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

Schritte zur Lösung

\int (5 x^{3} + 6 x) e^{x^{2}} d x

Multipliziere aus

(5 x^{3} + 6 x) e^{x^{2}} : 5 e^{x^{2}} x^{3} + 6 e^{x^{2}} x

= \int 5 e^{x^{2}} x^{3} + 6 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 5 e^{x^{2}} x^{3} d x + \int 6 e^{x^{2}} x d x

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

\int 6 e^{x^{2}} x d x = 3 e^{x^{2}}

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

Vereinfache

\frac{5}{2} (e^{x^{2}} x^{2} - e^{x^{2}}) + 3 e^{x^{2}} : \frac{5}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}}

= \frac{5}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{5}{2} e^{x^{2}} x^{2} + \frac{1}{2} e^{x^{2}} + C

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