integral from 0 to 4 of e^{-2x}

Lösung

\int_{0}^{4} e^{- 2 x} d x

\frac{e ^{8} - 1}{2 e ^{8}}

Dezimale

0.49983 \dots

Schritte zur Lösung

\int_{0}^{4} e^{- 2 x} d x

Wende U-Substitution an

= \int_{0}^{- 8} - \frac{1}{2} e^{u} d u

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

= - \int_{- 8}^{0} - \frac{1}{2} e^{u} d u

Entferne die Konstante:

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

= - (- \frac{1}{2} \cdot \int_{- 8}^{0} e^{u} d u)

Nutze das gemeinsame Integral :

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

= - (- \frac{1}{2} [e^{u}]_{- 8}^{0})

Vereinfache

= \frac{1}{2} [e^{u}]_{- 8}^{0}

Berechne die Grenzen:

1 - \frac{1}{e ^{8}}

= \frac{1}{2} (1 - \frac{1}{e ^{8}})

Vereinfache

= \frac{e ^{8} - 1}{2 e ^{8}}

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