integral from 0 to 2 of 9xe^{-3x}

Lösung

\int_{0}^{2} 9 x e^{- 3 x} d x

1 - \frac{7}{e ^{6}}

Dezimale

0.98264 \dots

Schritte zur Lösung

\int_{0}^{2} 9 x e^{- 3 x} d x

Entferne die Konstante:

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

= 9 \cdot \int_{0}^{2} x e^{- 3 x} d x

Wende U-Substitution an

= 9 \cdot \int_{0}^{- 6} \frac{e ^{u} u}{9} d u

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

= 9 (- \int_{- 6}^{0} \frac{e ^{u} u}{9} d u)

Entferne die Konstante:

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

= 9 (- \frac{1}{9} \cdot \int_{- 6}^{0} e^{u} u d u)

Wende die partielle Integration an

= 9 (- \frac{1}{9} [e^{u} u - \int e^{u} d u]_{- 6}^{0})

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

= 9 (- \frac{1}{9} [e^{u} u - e^{u}]_{- 6}^{0})

Vereinfache

9 (- \frac{1}{9} [e^{u} u - e^{u}]_{- 6}^{0}) : - [e^{u} u - e^{u}]_{- 6}^{0}

= - [e^{u} u - e^{u}]_{- 6}^{0}

Berechne die Grenzen:

- 1 + \frac{7}{e ^{6}}

= - (- 1 + \frac{7}{e ^{6}})

Vereinfache

= 1 - \frac{7}{e ^{6}}

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