integral (2x-x^2+7)e^2x

Lösung

integral

(2 x - x^{2} + 7) e^{2} x

e^{2} (\frac{2 x ^{3}}{3} - \frac{x ^{4}}{4} + \frac{7 x ^{2}}{2}) + C

Schritte zur Lösung

\int (2 x - x^{2} + 7) e^{2} x d x

Entferne die Konstante:

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

= e^{2} \cdot \int (2 x - x^{2} + 7) x d x

Multipliziere aus

(2 x - x^{2} + 7) x : 2 x^{2} - x^{3} + 7 x

= e^{2} \cdot \int 2 x^{2} - x^{3} + 7 x d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= e^{2} (\int 2 x^{2} d x - \int x^{3} d x + \int 7 x d x)

\int 2 x^{2} d x = \frac{2 x ^{3}}{3}

\int x^{3} d x = \frac{x ^{4}}{4}

\int 7 x d x = \frac{7 x ^{2}}{2}

= e^{2} (\frac{2 x ^{3}}{3} - \frac{x ^{4}}{4} + \frac{7 x ^{2}}{2})

Füge eine Konstante zur Lösung hinzu

= e^{2} (\frac{2 x ^{3}}{3} - \frac{x ^{4}}{4} + \frac{7 x ^{2}}{2}) + C

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