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Beliebt Rechnen >

integral of (x^2+3)e^{2x-3}

Lösung

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

Lösung

\frac{1}{4} e^{2 x - 3} (2 x^{2} - 2 x + 7) + C

Schritte zur Lösung

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

(x^{2} + 3) e^{2 x - 3} = e^{- 3} (x^{2} + 3) e^{2 x}

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

Entferne die Konstante:

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

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

Wende Exponentenregel an:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{e ^{3}} \cdot \int (x^{2} + 3) e^{2 x} d x

Wende die partielle Integration an

= \frac{1}{e ^{3}} (\frac{1}{2} e^{2 x} (x^{2} + 3) - \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}{e ^{3}} (\frac{1}{2} e^{2 x} (x^{2} + 3) - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))

Vereinfache

\frac{1}{e ^{3}} (\frac{1}{2} e^{2 x} (x^{2} + 3) - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})) : \frac{1}{4} e^{2 x - 3} (2 x^{2} - 2 x + 7)

= \frac{1}{4} e^{2 x - 3} (2 x^{2} - 2 x + 7)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} e^{2 x - 3} (2 x^{2} - 2 x + 7) + C

Graph

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