integral from 1 to 7 of e^{x^{-3}}

Lösung

\int_{1}^{7} e^{x^{- 3}} d x

- - \frac{1}{3} - \frac{3 e ^{1}}{3 1} - - 3 \cdot 3 \cdot \frac{1}{2} - \frac{( 3 1 ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 1 ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 1 ) ^{2} ) ^{\frac{3}{2}}} - - \frac{3 e ^{\frac{1}{343}}}{3 \frac{1}{343}} - - 3 \cdot 3 \cdot \frac{1}{2} - \frac{( 3 \frac{1}{343} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 \frac{1}{343} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 \frac{1}{343} ) ^{2} ) ^{\frac{3}{2}}}

Schritte zur Lösung

\int_{1}^{7} e^{x^{- 3}} d x

Wende U-Substitution an

= \int_{1}^{\frac{1}{343}} - \frac{e ^{u}}{3 u ^{\frac{4}{3}}} d u

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

= - \int_{\frac{1}{343}}^{1} - \frac{e ^{u}}{3 u ^{\frac{4}{3}}} d u

Entferne die Konstante:

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

= - (- \frac{1}{3} \cdot \int_{\frac{1}{343}}^{1} \frac{e ^{u}}{u ^{\frac{4}{3}}} d u)

Wende die partielle Integration an

= - (- \frac{1}{3} [- \frac{3 e ^{u}}{3 u} - \int - \frac{3 e ^{u}}{3 u} d u]_{\frac{1}{343}}^{1})

\int - \frac{3 e ^{u}}{3 u} d u = - 3 \cdot 3 \cdot \frac{1}{2} - \frac{( 3 u ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 u ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 u ) ^{2} ) ^{\frac{3}{2}}}

= - - \frac{1}{3} - \frac{3 e ^{u}}{3 u} - - 3 \cdot 3 \cdot \frac{1}{2} - \frac{( 3 u ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 u ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 u ) ^{2} ) ^{\frac{3}{2}}}_{\frac{1}{343}}^{1}

Berechne die Grenzen:

- \frac{3 e ^{1}}{3 1} - - 3 \cdot 3 \cdot \frac{1}{2} - \frac{( 3 1 ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 1 ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 1 ) ^{2} ) ^{\frac{3}{2}}} - - \frac{3 e ^{\frac{1}{343}}}{3 \frac{1}{343}} - - 3 \cdot 3 \cdot \frac{1}{2} - \frac{( 3 \frac{1}{343} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 \frac{1}{343} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 \frac{1}{343} ) ^{2} ) ^{\frac{3}{2}}}

= - - \frac{1}{3} - \frac{3 e ^{1}}{3 1} - - 3 \cdot 3 \cdot \frac{1}{2} - \frac{( 3 1 ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 1 ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 1 ) ^{2} ) ^{\frac{3}{2}}} - - \frac{3 e ^{\frac{1}{343}}}{3 \frac{1}{343}} - - 3 \cdot 3 \cdot \frac{1}{2} - \frac{( 3 \frac{1}{343} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 \frac{1}{343} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 \frac{1}{343} ) ^{2} ) ^{\frac{3}{2}}}

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