integral of x(b-x)^2

Lösung

\int x (b - x)^{2} d x

\frac{1}{2} b^{2} x^{2} - b x^{3} + \frac{1}{2} x^{4} + \frac{b x ^{3}}{3} - \frac{x ^{4}}{4} + C

Schritte zur Lösung

\int x (b - x)^{2} d x

Wende die partielle Integration an

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

\int - x^{2} (b - x) d x = - \frac{b x ^{3}}{3} + \frac{x ^{4}}{4}

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

Vereinfache

\frac{1}{2} x^{2} (b - x)^{2} - (- \frac{b x ^{3}}{3} + \frac{x ^{4}}{4}) : \frac{1}{2} b^{2} x^{2} - b x^{3} + \frac{1}{2} x^{4} + \frac{b x ^{3}}{3} - \frac{x ^{4}}{4}

= \frac{1}{2} b^{2} x^{2} - b x^{3} + \frac{1}{2} x^{4} + \frac{b x ^{3}}{3} - \frac{x ^{4}}{4}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} b^{2} x^{2} - b x^{3} + \frac{1}{2} x^{4} + \frac{b x ^{3}}{3} - \frac{x ^{4}}{4} + C

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