integral from 0 to 1 of xe^{-10x}

Lösung

\int_{0}^{1} x e^{- 10 x} d x

- \frac{- e ^{10} + 11}{100 e ^{10}}

Dezimale

0.00999 \dots

Schritte zur Lösung

\int_{0}^{1} x e^{- 10 x} d x

Wende U-Substitution an

= \int_{0}^{- 10} \frac{e ^{u} u}{100} d u

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

= - \int_{- 10}^{0} \frac{e ^{u} u}{100} d u

Entferne die Konstante:

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

= - \frac{1}{100} \cdot \int_{- 10}^{0} e^{u} u d u

Wende die partielle Integration an

= - \frac{1}{100} [e^{u} u - \int e^{u} d u]_{- 10}^{0}

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

= - \frac{1}{100} [e^{u} u - e^{u}]_{- 10}^{0}

Berechne die Grenzen:

- 1 + \frac{11}{e ^{10}}

= - \frac{1}{100} (- 1 + \frac{11}{e ^{10}})

Vereinfache

= - \frac{- e ^{10} + 11}{100 e ^{10}}