integral of x^2ye^{xy}

Lösung

\int x^{2} y e^{x y} d x

e^{y x} x^{2} - e^{y x} \frac{2}{y} x + e^{y x} \frac{2}{y ^{2}} + C

Schritte zur Lösung

\int x^{2} y e^{x y} d x

Entferne die Konstante:

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

= y \cdot \int e^{y x} x^{2} d x

Wende die partielle Integration an

= y (e^{y x} \frac{1}{y} x^{2} - \int e^{y x} \frac{2}{y} x d x)

\int e^{y x} \frac{2}{y} x d x = \frac{2}{y} (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}})

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

Vereinfache

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

= e^{y x} x^{2} - e^{y x} \frac{2}{y} x + e^{y x} \frac{2}{y ^{2}}

Füge eine Konstante zur Lösung hinzu

= e^{y x} x^{2} - e^{y x} \frac{2}{y} x + e^{y x} \frac{2}{y ^{2}} + C

(y - x 3) d x + (x + y 3) d y = 0

f^{^{'}} (x) = x e^{x^{2}}

\frac{\partial}{\partial x} (x + 6 y)

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