integral of 7x^3e^{5x}

Lösung

\int 7 x^{3} e^{5 x} d x

\frac{7}{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 7 x^{3} e^{5 x} d x

Entferne die Konstante:

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

= 7 \cdot \int x^{3} e^{5 x} d x

Wende U-Substitution an

= 7 \cdot \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

= 7 \cdot \frac{1}{625} \cdot \int e^{u} u^{3} d u

Wende die partielle Integration an

= 7 \cdot \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}))

= 7 \cdot \frac{1}{625} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Setze in

u = 5 x

ein

= 7 \cdot \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

7 \cdot \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{7}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

= \frac{7}{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{7}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) + C

t a y l or (1 + x)^{2}

x (\frac{d y}{d x}) - y = x^{2}

n = 0 \sum \infty \frac{2 ^{n}}{5 ^{n}}

x \to \infty - lim (\frac{1}{x})

x y^{^{'}} = x e^{- 2 x} - (2 x + 1) y