integral from 0 to 1 of pi(xe^x)^2

Lösung

\int_{0}^{1} π (x e^{x})^{2} d x

\frac{π ( e ^{2} - 1 )}{4}

Dezimale

5.01795 \dots

Schritte zur Lösung

\int_{0}^{1} π (x e^{x})^{2} d x

Entferne die Konstante:

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

= π \cdot \int_{0}^{1} (x e^{x})^{2} d x

Vereinfache

(x e^{x})^{2} : e^{2 x} x^{2}

= π \cdot \int_{0}^{1} e^{2 x} x^{2} d x

Wende die partielle Integration an

= π [\frac{1}{2} e^{2 x} x^{2} - \int e^{2 x} x d x]_{0}^{1}

\int e^{2 x} x d x = \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})

= π [\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})]_{0}^{1}

Berechne die Grenzen:

\frac{e ^{2}}{4} - \frac{1}{4}

= π (\frac{e ^{2}}{4} - \frac{1}{4})

Vereinfache

= \frac{π ( e ^{2} - 1 )}{4}

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