integral from 0 to 12 of 8xe^{-0.2x}

Lösung

\int_{0}^{12} 8 x e^{- 0.2 x} d x

138.31179 \dots

Schritte zur Lösung

\int_{0}^{12} 8 x e^{- 0.2 x} d x

Entferne die Konstante:

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

= 8 \cdot \int_{0}^{12} x e^{- 0.2 x} d x

Wende U-Substitution an

= 8 \cdot \int_{0}^{- 2.4} \frac{5}{0.2} e^{u} u d u

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

= 8 (- \int_{- 2.4}^{0} \frac{5}{0.2} e^{u} u d u)

Entferne die Konstante:

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

= 8 (- \frac{5}{0.2} \cdot \int_{- 2.4}^{0} e^{u} u d u)

Vereinfache

= 8 (- 25 \cdot \int_{- 2.4}^{0} e^{u} u d u)

Wende die partielle Integration an

= 8 (- 25 [e^{u} u - \int e^{u} d u]_{- 2.4}^{0})

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

= 8 (- 25 [e^{u} u - e^{u}]_{- 2.4}^{0})

Vereinfache

= - 200 [e^{u} u - e^{u}]_{- 2.4}^{0}

Berechne die Grenzen:

- 0.69155 \dots

= - 200 (- 0.69155 \dots)

Vereinfache

= 138.31179 \dots

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