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Beliebt Rechnen >

integral of (x-1)^2(x+2)

Lösung

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

Lösung

\frac{x ^{4}}{3} - x^{3} + \frac{2 x ^{3}}{3} - x^{2} + 2 x - \frac{x ^{4}}{12} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int \frac{x ^{3}}{3} - x^{2} + x d x = \frac{x ^{4}}{12} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2}

= (x + 2) (\frac{x ^{3}}{3} - x^{2} + x) - (\frac{x ^{4}}{12} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2})

Vereinfache

(x + 2) (\frac{x ^{3}}{3} - x^{2} + x) - (\frac{x ^{4}}{12} - \frac{x ^{3}}{3} + \frac{x ^{2}}{2}) : \frac{x ^{4}}{3} - x^{3} + \frac{2 x ^{3}}{3} - x^{2} + 2 x - \frac{x ^{4}}{12} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2}

= \frac{x ^{4}}{3} - x^{3} + \frac{2 x ^{3}}{3} - x^{2} + 2 x - \frac{x ^{4}}{12} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{4}}{3} - x^{3} + \frac{2 x ^{3}}{3} - x^{2} + 2 x - \frac{x ^{4}}{12} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + C

Graph

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