integral from 0 to x^3 of ye^{(-y)/x}

Lösung

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

- x^{4} e^{- x^{2}} - x^{2} e^{- x^{2}} + x^{2}

Schritte zur Lösung

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

Vereinfache

= \int_{0}^{x^{3}} e^{- \frac{y}{x}} y d y

Wende die partielle Integration an

= [- x e^{- \frac{y}{x}} y - \int - x e^{- \frac{y}{x}} d y]_{0}^{x^{3}}

\int - x e^{- \frac{y}{x}} d y = x^{2} e^{- \frac{y}{x}}

= [- x e^{- \frac{y}{x}} y - x^{2} e^{- \frac{y}{x}}]_{0}^{x^{3}}

Berechne die Grenzen:

- x^{4} e^{- x^{2}} - x^{2} e^{- x^{2}} + x^{2}

= - x^{4} e^{- x^{2}} - x^{2} e^{- x^{2}} + x^{2}

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