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Beliebt Rechnen >

integral from 0 to 2 of 7e^{-3x}

Lösung

\int_{0}^{2} 7 e^{- 3 x} d x

Lösung

\frac{7 ( e ^{6} - 1 )}{3 e ^{6}}

+1

Dezimale

2.32754 \dots

Schritte zur Lösung

\int_{0}^{2} 7 e^{- 3 x} d x

Entferne die Konstante:

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

= 7 \cdot \int_{0}^{2} e^{- 3 x} d x

Wende U-Substitution an

= 7 \cdot \int_{0}^{- 6} - \frac{1}{3} e^{u} d u

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

= 7 (- \int_{- 6}^{0} - \frac{1}{3} e^{u} d u)

Entferne die Konstante:

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

= 7 (- (- \frac{1}{3} \cdot \int_{- 6}^{0} e^{u} d u))

Nutze das gemeinsame Integral :

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

= 7 (- (- \frac{1}{3} [e^{u}]_{- 6}^{0}))

Vereinfache

7 (- (- \frac{1}{3} [e^{u}]_{- 6}^{0})) : \frac{7}{3} [e^{u}]_{- 6}^{0}

= \frac{7}{3} [e^{u}]_{- 6}^{0}

Berechne die Grenzen:

1 - \frac{1}{e ^{6}}

= \frac{7}{3} (1 - \frac{1}{e ^{6}})

Vereinfache

= \frac{7 ( e ^{6} - 1 )}{3 e ^{6}}

Graph

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