integral of (1+2x^2)e^{x^2}

Lösung

\int (1 + 2 x^{2}) e^{x^{2}} d x

e^{x^{2}} x + C

Schritte zur Lösung

\int (1 + 2 x^{2}) e^{x^{2}} d x

Multipliziere aus

(1 + 2 x^{2}) e^{x^{2}} : e^{x^{2}} + 2 e^{x^{2}} x^{2}

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

\int e^{x^{2}} d x = \frac{π}{2} erfi (x)

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

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

Vereinfache

\frac{π}{2} erfi (x) + e^{x^{2}} x - \frac{π}{2} erfi (x) : e^{x^{2}} x

= e^{x^{2}} x

Füge eine Konstante zur Lösung hinzu

= e^{x^{2}} x + C