integral from 0 to 1 of e^{-0.2x}

Lösung

\int_{0}^{1} e^{- 0.2 x} d x

0.90634 \dots

Schritte zur Lösung

\int_{0}^{1} e^{- 0.2 x} d x

Wende U-Substitution an

= \int_{0}^{- 0.2} - \frac{1}{0.2} e^{u} d u

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

= - \int_{- 0.2}^{0} - \frac{1}{0.2} e^{u} d u

Entferne die Konstante:

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

= - (- \frac{1}{0.2} \cdot \int_{- 0.2}^{0} e^{u} d u)

Vereinfache

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

Nutze das gemeinsame Integral :

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

= - (- 5 [e^{u}]_{- 0.2}^{0})

Vereinfache

= 5 [e^{u}]_{- 0.2}^{0}

Berechne die Grenzen:

0.18126 \dots

= 5 \cdot 0.18126 \dots

Vereinfache

= 0.90634 \dots

x \to \infty + lim (cos (x))

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