integral of 2e^{x/2}x^3

Lösung

\int 2 e^{\frac{x}{2}} x^{3} d x

2 (2 e^{\frac{x}{2}} x^{3} - 6 (2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}))) + C

Schritte zur Lösung

\int 2 e^{\frac{x}{2}} x^{3} d x

Entferne die Konstante:

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

= 2 \cdot \int e^{\frac{x}{2}} x^{3} d x

Wende die partielle Integration an

= 2 (2 e^{\frac{x}{2}} x^{3} - \int 6 e^{\frac{x}{2}} x^{2} d x)

\int 6 e^{\frac{x}{2}} x^{2} d x = 6 (2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}))

= 2 (2 e^{\frac{x}{2}} x^{3} - 6 (2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}})))

Füge eine Konstante zur Lösung hinzu

= 2 (2 e^{\frac{x}{2}} x^{3} - 6 (2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}))) + C

\frac{d}{d x} (\frac{3 - 5 x - x ^{2}}{x ^{2} - 6})

x \to 0 lim (\frac{x}{x ^{2} - 5 x})

x \to \infty + lim (x^{2})

s l o p e y = x^{2} - 2 x - 4

d er i v a t i v e cos (x^{3})