integral from 3 to sqrt(10 of)xe^{-x^2}

Lösung

\int_{3}^{10} x e^{- x^{2}} d x

\frac{e - 1}{2 e ^{10}}

Dezimale

0.00003 \dots

Schritte zur Lösung

\int_{3}^{10} x e^{- x^{2}} d x

Wende U-Substitution an

= \int_{- 9}^{- 10} - \frac{e ^{u}}{2} d u

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

= - \int_{- 10}^{- 9} - \frac{e ^{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_{- 10}^{- 9} e^{u} d u)

Nutze das gemeinsame Integral :

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

= - (- \frac{1}{2} [e^{u}]_{- 10}^{- 9})

Vereinfache

= \frac{1}{2} [e^{u}]_{- 10}^{- 9}

Berechne die Grenzen:

\frac{1}{e ^{9}} - \frac{1}{e ^{10}}

= \frac{1}{2} (\frac{1}{e ^{9}} - \frac{1}{e ^{10}})

Vereinfache

= \frac{e - 1}{2 e ^{10}}

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