integral of e^{(x^2)/2}*(-x^3)

Lösung

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

- e^{\frac{x ^{2}}{2}} x^{2} + 2 e^{\frac{x ^{2}}{2}} + C

Schritte zur Lösung

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

Vereinfache

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

Entferne die Konstante:

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

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

Wende U-Substitution an

= - \int \frac{e ^{\frac{u}{2}} u}{2} d u

Entferne die Konstante:

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

= - \frac{1}{2} \cdot \int e^{\frac{u}{2}} u d u

Wende U-Substitution an

= - \frac{1}{2} \cdot \int 4 e^{v} v d v

Entferne die Konstante:

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

= - \frac{1}{2} \cdot 4 \cdot \int e^{v} v d v

Wende die partielle Integration an

= - \frac{1}{2} \cdot 4 (e^{v} v - \int e^{v} d v)

\int e^{v} d v = e^{v}

= - \frac{1}{2} \cdot 4 (e^{v} v - e^{v})

Ersetze zurück

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

Vereinfache

- \frac{1}{2} \cdot 4 (e^{\frac{x ^{2}}{2}} \frac{x ^{2}}{2} - e^{\frac{x ^{2}}{2}}) : - e^{\frac{x ^{2}}{2}} x^{2} + 2 e^{\frac{x ^{2}}{2}}

= - e^{\frac{x ^{2}}{2}} x^{2} + 2 e^{\frac{x ^{2}}{2}}

Füge eine Konstante zur Lösung hinzu

= - e^{\frac{x ^{2}}{2}} x^{2} + 2 e^{\frac{x ^{2}}{2}} + C

\int 3 7^{5 x} d x

\int π (\frac{1}{x})^{2} d x

\int y e^{y} - e^{y} d y

\int \frac{2 - e ^{x}}{6 e ^{4 x}} d x

\int 1 + (\frac{2}{3} x^{- \frac{1}{3}})^{2} d x