integral from 0 to x^2 of x^2e^{x^2}

Lösung

\int_{0}^{x^{2}} x^{2} e^{x^{2}} d x

\frac{e ^{x^{4}} x ^{2}}{2} - \frac{π}{4} erfi (x^{2})

Schritte zur Lösung

\int_{0}^{x^{2}} x^{2} e^{x^{2}} d x

Vereinfache

x^{2} : xx

= \int_{0}^{xx} x^{2} e^{x^{2}} d x

Vereinfache

x^{2} : xx

= \int_{0}^{xx} xx e^{x^{2}} d x

Wende die partielle Integration an

= [\frac{1}{2} e^{x^{2}} x - \int \frac{1}{2} e^{x^{2}} d x]_{0}^{x^{2}}

\int \frac{1}{2} e^{x^{2}} d x = \frac{π}{4} erfi (x)

= [\frac{1}{2} e^{x^{2}} x - \frac{π}{4} erfi (x)]_{0}^{x^{2}}

Berechne die Grenzen:

\frac{1}{2} e^{x^{4}} x^{2} - \frac{π}{4} erfi (x^{2})

= \frac{1}{2} e^{x^{4}} x^{2} - \frac{π}{4} erfi (x^{2})

Vereinfache

= \frac{e ^{x^{4}} x ^{2}}{2} - \frac{π}{4} erfi (x^{2})