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

Lösung

\int_{0}^{3} - e^{- 2 y} d y

- \frac{e ^{6} - 1}{2 e ^{6}}

Dezimale

- 0.49876 \dots

Schritte zur Lösung

\int_{0}^{3} - e^{- 2 y} d y

Entferne die Konstante:

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

= - \int_{0}^{3} e^{- 2 y} d y

Wende U-Substitution an

= - \int_{0}^{- 6} - \frac{1}{2} e^{u} d u

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

= - (- \int_{- 6}^{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_{- 6}^{0} e^{u} d u))

Nutze das gemeinsame Integral :

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

= - (- (- \frac{1}{2} [e^{u}]_{- 6}^{0}))

Vereinfache

= - \frac{1}{2} [e^{u}]_{- 6}^{0}

Berechne die Grenzen:

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

= - \frac{1}{2} (1 - \frac{1}{e ^{6}})

Vereinfache

= - \frac{e ^{6} - 1}{2 e ^{6}}

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