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

Lösung

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

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

Dezimale

1.26424 \dots

Schritte zur Lösung

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

Wende U-Substitution an

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

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

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

Entferne die Konstante:

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

= - (- 2 \cdot \int_{- 1}^{0} e^{u} d u)

Nutze das gemeinsame Integral :

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

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

Vereinfache

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

Berechne die Grenzen:

1 - \frac{1}{e}

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

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