integral from 0 to 1 of 1/2 e^{-x/2}

Lösung

\int_{0}^{1} \frac{1}{2} e^{- \frac{x}{2}} d x

1 - \frac{1}{e}

Dezimale

0.39346 \dots

Schritte zur Lösung

\int_{0}^{1} \frac{1}{2} e^{- \frac{x}{2}} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int_{0}^{1} e^{- \frac{x}{2}} d x

Wende U-Substitution an

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

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

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

Nutze das gemeinsame Integral :

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

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

Vereinfache

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

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

Berechne die Grenzen:

1 - \frac{1}{e}

= 1 - \frac{1}{e}

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