integral x^2e^{(x^2)/2}

Lösung

integral

x^{2} e^{\frac{x ^{2}}{2}}

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

Schritte zur Lösung

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

Wende Exponentenregel an:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{x ^{2}}{2} = (\frac{x}{2})^{2}

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

Wende U-Substitution an

= \int 22 e^{u^{2}} u^{2} d u

Entferne die Konstante:

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

= 22 \cdot \int e^{u^{2}} u^{2} d u

Vereinfache

u^{2} : uu

= 22 \cdot \int e^{u^{2}} uu d u

Wende die partielle Integration an

= 22 (\frac{1}{2} e^{u^{2}} u - \int \frac{1}{2} e^{u^{2}} d u)

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

= 22 (\frac{1}{2} e^{u^{2}} u - \frac{π}{4} erfi (u))

Setze in

u = \frac{x}{2}

ein

= 22 (\frac{1}{2} e^{(\frac{x}{2})^{2}} \frac{x}{2} - \frac{π}{4} erfi (\frac{x}{2}))

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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