integral from 1/2 x to 1 of x^2e^{xy}

Lösung

\int_{\frac{1}{2} x}^{1} x^{2} e^{x y} d y

x (e^{x} - e^{\frac{x ^{2}}{2}})

Schritte zur Lösung

\int_{\frac{1}{2} x}^{1} x^{2} e^{x y} d y

Entferne die Konstante:

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

= x^{2} \cdot \int_{\frac{1}{2} x}^{1} e^{x y} d y

Wende U-Substitution an

= x^{2} \cdot \int_{\frac{x ^{2}}{2}}^{x} \frac{e ^{u}}{x} d u

Entferne die Konstante:

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

= x^{2} \frac{1}{x} \cdot \int_{\frac{x ^{2}}{2}}^{x} e^{u} d u

Nutze das gemeinsame Integral :

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

= x^{2} \frac{1}{x} [e^{u}]_{\frac{x ^{2}}{2}}^{x}

Vereinfache

x^{2} \frac{1}{x} [e^{u}]_{\frac{x ^{2}}{2}}^{x} : x [e^{u}]_{\frac{x ^{2}}{2}}^{x}

= x [e^{u}]_{\frac{x ^{2}}{2}}^{x}

Berechne die Grenzen:

e^{x} - e^{\frac{x ^{2}}{2}}

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

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