integral of x^3e^{xy}

Lösung

\int x^{3} e^{x y}

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{e ^{u} u ^{3}}{y ^{4}} d u

Entferne die Konstante:

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

= \frac{1}{y ^{4}} \cdot \int e^{u} u^{3} d u

Wende die partielle Integration an

= \frac{1}{y ^{4}} (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}{y ^{4}} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Setze in

u = y x

ein

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

Vereinfache

\frac{1}{y ^{4}} ((y x)^{3} e^{y x} - 3 ((y x)^{2} e^{y x} - 2 (e^{y x} y x - e^{y x}))) : \frac{1}{y ^{4}} (y^{3} e^{y x} x^{3} - 3 (y^{2} e^{y x} x^{2} - 2 (y e^{y x} x - e^{y x})))

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

Füge eine Konstante zur Lösung hinzu

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

\int_{6}^{8} \frac{44}{( x - 6 ) ^{3}} d x

\frac{d}{d x} (- \frac{10}{x ^{3}})

\int_{0}^{\frac{π}{4}} sec^{3} (x) d x

y^{^{'}} + y = cos (x)

t an g e n t f (x) = 4 x^{2} + 2 x - 1, at x = 2