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

Lösung

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

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

Dezimale

0.43233 \dots

Schritte zur Lösung

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

Wende Exponentenregel an:

\frac{1}{a ^{b}} = a^{- b}

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

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

Wende U-Substitution an

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

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

= - \int_{- 2}^{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_{- 2}^{0} e^{u} d u)

Nutze das gemeinsame Integral :

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

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

Vereinfache

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

Berechne die Grenzen:

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

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

Vereinfache

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

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