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

Lösung

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

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

Wende die partielle Integration an

= 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 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 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} ((3 x^{2} + 7)^{3})

x \to + 2 - lim (x (x - 1))

\int \frac{2 x ^{3}}{( x ^{2} + 1 ) ^{2}} d x

\frac{d}{d x} (\frac{( x + 1 )}{( x ^{3} + x - 2 )})

t an g e n t \frac{2 x}{x ^{2} + 1}