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Beliebt Rechnen >

integral from 0 to 1 of 2pix(3e^{-x^2})

Lösung

\int_{0}^{1} 2 π x (3 e^{- x^{2}}) d x

Lösung

3 π (1 - \frac{1}{e})

+1

Dezimale

5.95759 \dots

Schritte zur Lösung

\int_{0}^{1} 2 π x \cdot 3 e^{- x^{2}} d x

Entferne die Konstante:

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

= 2 π 3 \cdot \int_{0}^{1} x e^{- x^{2}} d x

Vereinfache

= 6 π \cdot \int_{0}^{1} x e^{- x^{2}} d x

Wende U-Substitution an

= 6 π \cdot \int_{0}^{- 1} - \frac{e ^{u}}{2} d u

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

= 6 π (- \int_{- 1}^{0} - \frac{e ^{u}}{2} d u)

Entferne die Konstante:

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

= 6 π (- (- \frac{1}{2} \cdot \int_{- 1}^{0} e^{u} d u))

Nutze das gemeinsame Integral :

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

= 6 π (- (- \frac{1}{2} [e^{u}]_{- 1}^{0}))

Vereinfache

6 π (- (- \frac{1}{2} [e^{u}]_{- 1}^{0})) : 3 π [e^{u}]_{- 1}^{0}

= 3 π [e^{u}]_{- 1}^{0}

Berechne die Grenzen:

1 - \frac{1}{e}

= 3 π (1 - \frac{1}{e})

Graph

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