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

Lösung

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

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

Dezimale

0.43233 \dots

Schritte zur Lösung

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

Wende U-Substitution an

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

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

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

Entferne die Konstante:

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

= - (- \int_{\frac{1}{e}}^{1} u d u)

Wende die Potenzregel an

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

Vereinfache

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

Berechne die Grenzen:

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

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

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