integral from 0 to 2 of 5xe^{-x^2}

Lösung

\int_{0}^{2} 5 x e^{- x^{2}} d x

\frac{5 ( e ^{4} - 1 )}{2 e ^{4}}

Dezimale

2.45421 \dots

Schritte zur Lösung

\int_{0}^{2} 5 x e^{- x^{2}} d x

Entferne die Konstante:

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

= 5 \cdot \int_{0}^{2} x e^{- x^{2}} d x

Wende U-Substitution an

= 5 \cdot \int_{0}^{- 4} - \frac{e ^{u}}{2} d u

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

= 5 (- \int_{- 4}^{0} - \frac{e ^{u}}{2} d u)

Entferne die Konstante:

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

= 5 (- (- \frac{1}{2} \cdot \int_{- 4}^{0} e^{u} d u))

Nutze das gemeinsame Integral :

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

= 5 (- (- \frac{1}{2} [e^{u}]_{- 4}^{0}))

Vereinfache

5 (- (- \frac{1}{2} [e^{u}]_{- 4}^{0})) : \frac{5}{2} [e^{u}]_{- 4}^{0}

= \frac{5}{2} [e^{u}]_{- 4}^{0}

Berechne die Grenzen:

1 - \frac{1}{e ^{4}}

= \frac{5}{2} (1 - \frac{1}{e ^{4}})

Vereinfache

= \frac{5 ( e ^{4} - 1 )}{2 e ^{4}}

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