integral from 2 to 6 of e^{-x/2}

Lösung

\int_{2}^{6} e^{- \frac{x}{2}} d x

2 (\frac{1}{e} - \frac{1}{e ^{3}})

Dezimale

0.63618 \dots

Schritte zur Lösung

\int_{2}^{6} e^{- \frac{x}{2}} d x

Wende U-Substitution an

= \int_{- 1}^{- 3} - 2 e^{u} d u

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

= - \int_{- 3}^{- 1} - 2 e^{u} d u

Entferne die Konstante:

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

= - (- 2 \cdot \int_{- 3}^{- 1} e^{u} d u)

Nutze das gemeinsame Integral :

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

= - (- 2 [e^{u}]_{- 3}^{- 1})

Vereinfache

= 2 [e^{u}]_{- 3}^{- 1}

Berechne die Grenzen:

\frac{1}{e} - \frac{1}{e ^{3}}

= 2 (\frac{1}{e} - \frac{1}{e ^{3}})

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