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

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

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

Wende die partielle Integration an

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

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

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

Berechne die Grenzen:

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

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

\int - 2 - \frac{2}{x} d x

\frac{d y}{d x} = \frac{2 x ^{2}}{y}

\frac{d}{d x} (2 x^{\frac{2}{3}})

d er i v a t i v e 2 + 3 x

\frac{d}{d x} (- 9 sin (3 x))