integral of (e^{1/x})/(x^4)

Lösung

\int \frac{e ^{\frac{1}{x}}}{x ^{4}} d x

- \frac{e ^{\frac{1}{x}}}{x ^{2}} + \frac{2 e ^{\frac{1}{x}}}{x} - 2 e^{\frac{1}{x}} + C

Schritte zur Lösung

\int \frac{e ^{\frac{1}{x}}}{x ^{4}} d x

Wende U-Substitution an

= \int - e^{u} u^{2} d u

Entferne die Konstante:

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

= - \int e^{u} u^{2} d u

Wende die partielle Integration an

= - (u^{2} e^{u} - \int 2 u e^{u} d u)

\int 2 u e^{u} d u = 2 (e^{u} u - e^{u})

= - (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Setze in

u = \frac{1}{x}

ein

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

Vereinfache

- ((\frac{1}{x})^{2} e^{\frac{1}{x}} - 2 (e^{\frac{1}{x}} \frac{1}{x} - e^{\frac{1}{x}})) : - \frac{e ^{\frac{1}{x}}}{x ^{2}} + \frac{2 e ^{\frac{1}{x}}}{x} - 2 e^{\frac{1}{x}}

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

Füge eine Konstante zur Lösung hinzu

= - \frac{e ^{\frac{1}{x}}}{x ^{2}} + \frac{2 e ^{\frac{1}{x}}}{x} - 2 e^{\frac{1}{x}} + C

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