integral from 0 to 4 of xe^{-3x}

Lösung

\int_{0}^{4} x e^{- 3 x} d x

- \frac{- e ^{12} + 13}{9 e ^{12}}

Dezimale

0.11110 \dots

Schritte zur Lösung

\int_{0}^{4} x e^{- 3 x} d x

Wende U-Substitution an

= \int_{0}^{- 12} \frac{e ^{u} u}{9} d u

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

= - \int_{- 12}^{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

= - \frac{1}{9} \cdot \int_{- 12}^{0} e^{u} u d u

Wende die partielle Integration an

= - \frac{1}{9} [e^{u} u - \int e^{u} d u]_{- 12}^{0}

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

= - \frac{1}{9} [e^{u} u - e^{u}]_{- 12}^{0}

Berechne die Grenzen:

- 1 + \frac{13}{e ^{12}}

= - \frac{1}{9} (- 1 + \frac{13}{e ^{12}})

Vereinfache

= - \frac{- e ^{12} + 13}{9 e ^{12}}

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