integral from 1 to 2 of (e^{1/x})/(x^3)

Lösung

\int_{1}^{2} \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

\frac{e}{2}

Dezimale

0.82436 \dots

Schritte zur Lösung

\int_{1}^{2} \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

Wende U-Substitution an

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

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

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Berechne die Grenzen:

\frac{e}{2}

= \frac{e}{2}

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