integral from-1 to 0 of 5(e^{1/x})/(x^3)

Lösung

\int_{- 1}^{0} 5 \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

- \frac{10}{e}

Dezimale

- 3.67879 \dots

Schritte zur Lösung

\int_{- 1}^{0} 5 \cdot \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

Entferne die Konstante:

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

= 5 \cdot \int_{- 1}^{0} \frac{e ^{\frac{1}{x}}}{x ^{3}} d x

Wende die partielle Integration an

= 5 [\frac{e ^{\frac{1}{x}}}{x ^{2}} - \int - \frac{e ^{\frac{1}{x}} ( 3 x + 1 )}{x ^{4}} d x]_{- 1}^{0}

\int - \frac{e ^{\frac{1}{x}} ( 3 x + 1 )}{x ^{4}} d x = \frac{3 e ^{\frac{1}{x}}}{x} - e^{\frac{1}{x}} + \frac{e ^{\frac{1}{x}}}{x ^{2}} - \frac{2 e ^{\frac{1}{x}}}{x}

= 5 [\frac{e ^{\frac{1}{x}}}{x ^{2}} - (\frac{3 e ^{\frac{1}{x}}}{x} - e^{\frac{1}{x}} + \frac{e ^{\frac{1}{x}}}{x ^{2}} - \frac{2 e ^{\frac{1}{x}}}{x})]_{- 1}^{0}

Vereinfache

5 [\frac{e ^{\frac{1}{x}}}{x ^{2}} - (\frac{3 e ^{\frac{1}{x}}}{x} - e^{\frac{1}{x}} + \frac{e ^{\frac{1}{x}}}{x ^{2}} - \frac{2 e ^{\frac{1}{x}}}{x})]_{- 1}^{0} : 5 [\frac{- e ^{\frac{1}{x}} + e ^{\frac{1}{x}} x}{x}]_{- 1}^{0}

= 5 [\frac{- e ^{\frac{1}{x}} + e ^{\frac{1}{x}} x}{x}]_{- 1}^{0}

Berechne die Grenzen:

- \frac{2}{e}

= 5 (- \frac{2}{e})

Vereinfache

= - \frac{10}{e}

\int_{1}^{4} \frac{1}{x + 1 + x} d x

\int_{- 5}^{- 1} \frac{1}{16} d x

\int_{0}^{\frac{π}{2}} e^{- s t} sin (t) d t

\int_{0}^{5} (x^{2} + 4) d x

\int_{1}^{2} (x - 4) d x