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Beliebt Rechnen >

integral of (x^2-1)e^{2x}

Lösung

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

Lösung

\frac{1}{2} e^{2 x} x^{2} - \frac{1}{2} e^{2 x} x - \frac{1}{4} e^{2 x} + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

\frac{1}{2} e^{2 x} (x^{2} - 1) - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}) : \frac{1}{2} e^{2 x} x^{2} - \frac{1}{2} e^{2 x} x - \frac{1}{4} e^{2 x}

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} e^{2 x} x^{2} - \frac{1}{2} e^{2 x} x - \frac{1}{4} e^{2 x} + C

Graph

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