integral of x^3e^{5x}

Lösung

\int x^{3} e^{5 x} d x

\frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) + C

Schritte zur Lösung

\int x^{3} e^{5 x} d x

Wende U-Substitution an

= \int \frac{e ^{u} u ^{3}}{625} d u

Entferne die Konstante:

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

= \frac{1}{625} \cdot \int e^{u} u^{3} d u

Wende die partielle Integration an

= \frac{1}{625} (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

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

= \frac{1}{625} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Setze in

u = 5 x

ein

= \frac{1}{625} ((5 x)^{3} e^{5 x} - 3 ((5 x)^{2} e^{5 x} - 2 (e^{5 x} \cdot 5 x - e^{5 x})))

Vereinfache

\frac{1}{625} ((5 x)^{3} e^{5 x} - 3 ((5 x)^{2} e^{5 x} - 2 (e^{5 x} \cdot 5 x - e^{5 x}))) : \frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

= \frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) + C

l a pl a ce t r an s f or m 1 - 2 e^{- 2 t} + e^{- 4 t}

\int_{\frac{1}{2}}^{1} (x^{- 3} - 17) d x

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

d er i v a t i v e 14 x^{\frac{1}{4}}

x \to 0 lim (f (x) (x + 1))