integral from 0 to 4 of xe^{(-1/2)x^2}

Lösung

\int_{0}^{4} x e^{(- \frac{1}{2}) x^{2}} d x

1 - \frac{1}{e ^{8}}

Dezimale

0.99966 \dots

Schritte zur Lösung

\int_{0}^{4} x e^{(- \frac{1}{2}) x^{2}} d x

Faktorisiere

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

= \int_{0}^{4} e^{- \frac{x ^{2}}{2}} x d x

Wende U-Substitution an

= \int_{0}^{- 8} - e^{u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 8}^{0} - e^{u} d u

Entferne die Konstante:

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

= - (- \int_{- 8}^{0} e^{u} d u)

Nutze das gemeinsame Integral :

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

= - (- [e^{u}]_{- 8}^{0})

Vereinfache

= [e^{u}]_{- 8}^{0}

Berechne die Grenzen:

1 - \frac{1}{e ^{8}}

= 1 - \frac{1}{e ^{8}}