integral from 0 to 1 of x^2e^{xy}

Lösung

\int_{0}^{1} x^{2} e^{x y} d y

x (e^{x} - 1)

Schritte zur Lösung

\int_{0}^{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_{0}^{1} e^{x y} d y

Wende U-Substitution an

= x^{2} \cdot \int_{0}^{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_{0}^{x} e^{u} d u

Nutze das gemeinsame Integral :

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

= x^{2} \frac{1}{x} [e^{u}]_{0}^{x}

Vereinfache

x^{2} \frac{1}{x} [e^{u}]_{0}^{x} : x [e^{u}]_{0}^{x}

= x [e^{u}]_{0}^{x}

Berechne die Grenzen:

e^{x} - 1

= x (e^{x} - 1)

\int_{0}^{1} \frac{x}{2 - x ^{2}} d x

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