integral from 0 to 7 of e^{-4x}

Lösung

\int_{0}^{7} e^{- 4 x} d x

\frac{e ^{28} - 1}{4 e ^{28}}

Dezimale

0.24999 \dots

Schritte zur Lösung

\int_{0}^{7} e^{- 4 x} d x

Wende U-Substitution an

= \int_{0}^{- 28} - \frac{1}{4} e^{u} d u

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

= - \int_{- 28}^{0} - \frac{1}{4} e^{u} d u

Entferne die Konstante:

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

= - (- \frac{1}{4} \cdot \int_{- 28}^{0} e^{u} d u)

Nutze das gemeinsame Integral :

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

= - (- \frac{1}{4} [e^{u}]_{- 28}^{0})

Vereinfache

= \frac{1}{4} [e^{u}]_{- 28}^{0}

Berechne die Grenzen:

1 - \frac{1}{e ^{28}}

= \frac{1}{4} (1 - \frac{1}{e ^{28}})

Vereinfache

= \frac{e ^{28} - 1}{4 e ^{28}}