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

Lösung

\int_{1}^{2} (e^{(x - 1)})^{2} d x

\frac{e ^{4} - e ^{2}}{2 e ^{2}}

Dezimale

3.19452 \dots

Schritte zur Lösung

\int_{1}^{2} (e^{x - 1})^{2} d x

(e^{x - 1})^{2} = (e^{- 1})^{2} (e^{x})^{2}

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

Entferne die Konstante:

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

= (e^{- 1})^{2} \cdot \int_{1}^{2} (e^{x})^{2} d x

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

= \frac{1}{e ^{2}} \cdot \int_{1}^{2} (e^{x})^{2} d x

Wende U-Substitution an

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

Wende die Potenzregel an

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

Berechne die Grenzen:

\frac{e ^{4} - e ^{2}}{2}

= \frac{1}{e ^{2}} \cdot \frac{e ^{4} - e ^{2}}{2}

Vereinfache

= \frac{e ^{4} - e ^{2}}{2 e ^{2}}

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