integral from 0 to 2 of (x^2ye^{xy})

Lösung

\int_{0}^{2} (x^{2} y e^{x y}) d x

4 e^{2 y} + \frac{- 2 ( 2 y e ^{2 y} - e ^{2 y} ) - 2}{y ^{2}}

Schritte zur Lösung

\int_{0}^{2} 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_{0}^{2} 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]_{0}^{2}

\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}})]_{0}^{2}

Berechne die Grenzen:

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

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

Vereinfache

= 4 e^{2 y} + \frac{- 2 ( 2 y e ^{2 y} - e ^{2 y} ) - 2}{y ^{2}}

\int_{2}^{4} x^{3} + 4 d x

\int_{1}^{e^{10}} x ln (x) d x

\int_{x}^{\frac{π}{2}} \frac{sin ( x )}{x} d x

\int_{- 2}^{0} x^{2} - 4 d x

\int_{1}^{2} x^{4} ln (3 x) d x