integral of x^2e^{-(2x)/a}

Lösung

\int x^{2} e^{- \frac{2 x}{a}} d x

- \frac{a ( 2 e ^{- \frac{2 x}{a}} x ^{2} - a ^{2} ( - \frac{2 e ^{- \frac{2 x}{a}} x}{a} - e ^{- \frac{2 x}{a}} ) )}{4} + C

Schritte zur Lösung

\int x^{2} e^{- \frac{2 x}{a}} d x

Wende U-Substitution an

= \int - \frac{a ^{3} e ^{u} u ^{2}}{8} d u

Entferne die Konstante:

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

= - \frac{a ^{3}}{8} \cdot \int e^{u} u^{2} d u

Wende die partielle Integration an

= - \frac{a ^{3}}{8} (u^{2} e^{u} - \int 2 u e^{u} d u)

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

= - \frac{a ^{3}}{8} (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Setze in

u = - \frac{2 x}{a}

ein

= - \frac{a ^{3}}{8} ((- \frac{2 x}{a})^{2} e^{- \frac{2 x}{a}} - 2 (e^{- \frac{2 x}{a}} (- \frac{2 x}{a}) - e^{- \frac{2 x}{a}}))

Vereinfache

- \frac{a ^{3}}{8} ((- \frac{2 x}{a})^{2} e^{- \frac{2 x}{a}} - 2 (e^{- \frac{2 x}{a}} (- \frac{2 x}{a}) - e^{- \frac{2 x}{a}})) : - \frac{a ( 2 e ^{- \frac{2 x}{a}} x ^{2} - a ^{2} ( - \frac{2 e ^{- \frac{2 x}{a}} x}{a} - e ^{- \frac{2 x}{a}} ) )}{4}

= - \frac{a ( 2 e ^{- \frac{2 x}{a}} x ^{2} - a ^{2} ( - \frac{2 e ^{- \frac{2 x}{a}} x}{a} - e ^{- \frac{2 x}{a}} ) )}{4}

Füge eine Konstante zur Lösung hinzu

= - \frac{a ( 2 e ^{- \frac{2 x}{a}} x ^{2} - a ^{2} ( - \frac{2 e ^{- \frac{2 x}{a}} x}{a} - e ^{- \frac{2 x}{a}} ) )}{4} + C

x \to 0 lim (x \cdot cos (\frac{1}{x}))

t an g e n t 3 x^{2} - 5 x + 2

\frac{d}{d x} (\frac{x ^{2} + 5 x - 1}{x ^{2}})

d er i v a t i v e - 3 + 12 x - 3 x^{2}

s l o p e in t erce pt - 6 x = 10 - y - 36 x = 48 - 6 y