integral from 0 to 0.2 of 25e^{-5x}

Lösung

\int_{0}^{0.2} 25 e^{- 5 x} d x

5 (1 - \frac{1}{e})

Dezimale

3.16060 \dots

Schritte zur Lösung

\int_{0}^{0.2} 25 e^{- 5 x} d x

Entferne die Konstante:

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

= 25 \cdot \int_{0}^{0.2} e^{- 5 x} d x

Wende U-Substitution an

= 25 \cdot \int_{0}^{- 1} - \frac{1}{5} e^{u} d u

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

= 25 (- \int_{- 1}^{0} - \frac{1}{5} e^{u} d u)

Entferne die Konstante:

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

= 25 (- (- \frac{1}{5} \cdot \int_{- 1}^{0} e^{u} d u))

Nutze das gemeinsame Integral :

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

= 25 (- (- \frac{1}{5} [e^{u}]_{- 1}^{0}))

Vereinfache

25 (- (- \frac{1}{5} [e^{u}]_{- 1}^{0})) : 5 [e^{u}]_{- 1}^{0}

= 5 [e^{u}]_{- 1}^{0}

Berechne die Grenzen:

1 - \frac{1}{e}

= 5 (1 - \frac{1}{e})

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