integral from-1 to 5 of 5xe^{-x^2}

Lösung

\int_{- 1}^{5} 5 x e^{- x^{2}} d x

\frac{5 ( \frac{1}{e} - \frac{1}{e ^{25}} )}{2}

Dezimale

0.91969 \dots

Schritte zur Lösung

\int_{- 1}^{5} 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_{- 1}^{5} x e^{- x^{2}} d x

Wende U-Substitution an

= 5 \cdot \int_{- 1}^{- 25} - \frac{e ^{u}}{2} d u

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

= 5 (- \int_{- 25}^{- 1} - \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_{- 25}^{- 1} e^{u} d u))

Nutze das gemeinsame Integral :

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

= 5 (- (- \frac{1}{2} [e^{u}]_{- 25}^{- 1}))

Vereinfache

5 (- (- \frac{1}{2} [e^{u}]_{- 25}^{- 1})) : \frac{5}{2} [e^{u}]_{- 25}^{- 1}

= \frac{5}{2} [e^{u}]_{- 25}^{- 1}

Berechne die Grenzen:

\frac{1}{e} - \frac{1}{e ^{25}}

= \frac{5}{2} (\frac{1}{e} - \frac{1}{e ^{25}})

Vereinfache

= \frac{5 ( \frac{1}{e} - \frac{1}{e ^{25}} )}{2}

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