integral of-4x^3e^{4x^2}

Lösung

\int - 4 x^{3} e^{4 x^{2}} d x

- \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}}) + C

Schritte zur Lösung

\int - 4 x^{3} e^{4 x^{2}} d x

Entferne die Konstante:

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

= - 4 \cdot \int x^{3} e^{4 x^{2}} d x

Wende U-Substitution an

= - 4 \cdot \int \frac{e ^{u} u}{32} d u

Entferne die Konstante:

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

= - 4 \cdot \frac{1}{32} \cdot \int e^{u} u d u

Wende die partielle Integration an

= - 4 \cdot \frac{1}{32} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= - 4 \cdot \frac{1}{32} (e^{u} u - e^{u})

Setze in

u = 4 x^{2}

ein

= - 4 \cdot \frac{1}{32} (e^{4 x^{2}} \cdot 4 x^{2} - e^{4 x^{2}})

Vereinfache

- 4 \cdot \frac{1}{32} (e^{4 x^{2}} \cdot 4 x^{2} - e^{4 x^{2}}) : - \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}})

= - \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}})

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{8} (4 e^{4 x^{2}} x^{2} - e^{4 x^{2}}) + C

y^{^{'}} + y = y^{- 2}

\int e^{3 t} (e^{3 t} - 1) d t

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

s l o p e (- 1, 6), (7, - 2)

\frac{d y}{d x} = y - x