integral of x^3e^{4x}

Lösung

\int x^{3} e^{4 x} d x

\frac{1}{256} (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x}))) + C

Schritte zur Lösung

\int x^{3} e^{4 x} d x

Wende U-Substitution an

= \int \frac{e ^{u} u ^{3}}{256} d u

Entferne die Konstante:

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

= \frac{1}{256} \cdot \int e^{u} u^{3} d u

Wende die partielle Integration an

= \frac{1}{256} (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}{256} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Setze in

u = 4 x

ein

= \frac{1}{256} ((4 x)^{3} e^{4 x} - 3 ((4 x)^{2} e^{4 x} - 2 (e^{4 x} \cdot 4 x - e^{4 x})))

Vereinfache

\frac{1}{256} ((4 x)^{3} e^{4 x} - 3 ((4 x)^{2} e^{4 x} - 2 (e^{4 x} \cdot 4 x - e^{4 x}))) : \frac{1}{256} (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x})))

= \frac{1}{256} (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x})))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{256} (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x}))) + C

x \to 0 lim (sin^{3} (2 x))

\frac{d}{d x} (\frac{- 8 x}{( x ^{2} - 16 ) ^{2}})

\int tan^{5} (x) sec (x) d x

ma c l a u r in 5 e^{- x}

\frac{d}{d x} ((sin (x))^{l n (x)})