integral of x^3e^{12x}

Lösung

\int x^{3} e^{12 x} d x

\frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1) + C

Schritte zur Lösung

\int x^{3} e^{12 x} d x

Wende U-Substitution an

= \int \frac{e ^{u} u ^{3}}{20736} d u

Entferne die Konstante:

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

= \frac{1}{20736} \cdot \int e^{u} u^{3} d u

Wende die partielle Integration an

= \frac{1}{20736} (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}{20736} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Setze in

u = 12 x

ein

= \frac{1}{20736} ((12 x)^{3} e^{12 x} - 3 ((12 x)^{2} e^{12 x} - 2 (e^{12 x} \cdot 12 x - e^{12 x})))

Vereinfache

\frac{1}{20736} ((12 x)^{3} e^{12 x} - 3 ((12 x)^{2} e^{12 x} - 2 (e^{12 x} \cdot 12 x - e^{12 x}))) : \frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1)

= \frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3456} e^{12 x} (288 x^{3} - 72 x^{2} + 12 x - 1) + C

\frac{d}{d x} ((x^{4} + 3 x)^{- 1})

in v erse l a pl a ce \frac{s - 1}{( s + 1 ) ( s + 2 ) s}

f (x) = \frac{5}{x ^{5}}

t a y l or f (x) = sin (x), 0

a re a x^{2} + x + 3, [1, 3]