integral 1/(x^3*(e^{-1/x))}

Lösung

integral

\frac{1}{x ^{3} \cdot ( e ^{- \frac{1}{x}} )}

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

Schritte zur Lösung

\int \frac{1}{x ^{3} e ^{- \frac{1}{x}}} d x

Wende Exponentenregel an:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{x ^{3} e ^{- \frac{1}{x}}} = x^{- 3} \frac{1}{e ^{- \frac{1}{x}}}

= \int x^{- 3} \frac{1}{e ^{- \frac{1}{x}}} d x

Vereinfache

x^{- 3} \frac{1}{e ^{- \frac{1}{x}}} : \frac{e ^{\frac{1}{x}}}{x ^{3}}

= \int \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

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

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

Setze in

u = \frac{1}{x}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l \frac{1}{1 + e ^{- 2 x}}

in t e g r a l (a^{2} - x^{2})^{\frac{1}{2}}

in t e g r a l x^{3} + x + 1

in t e g r a l \frac{x ^{3}}{( x - 1 )}

in t e g r a l \frac{1}{( a - x ) ^{3}}