integral of x(5-x)^2

Lösung

\int x (5 - x)^{2} d x

\frac{25 x ^{2}}{2} - \frac{10 x ^{3}}{3} + \frac{x ^{4}}{4} + C

Schritte zur Lösung

\int x (5 - x)^{2} d x

Multipliziere aus

x (5 - x)^{2} : 25 x - 10 x^{2} + x^{3}

= \int 25 x - 10 x^{2} + x^{3} 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

= \int 25 x d x - \int 10 x^{2} d x + \int x^{3} d x

\int 25 x d x = \frac{25 x ^{2}}{2}

\int 10 x^{2} d x = \frac{10 x ^{3}}{3}

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

= \frac{25 x ^{2}}{2} - \frac{10 x ^{3}}{3} + \frac{x ^{4}}{4}

Füge eine Konstante zur Lösung hinzu

= \frac{25 x ^{2}}{2} - \frac{10 x ^{3}}{3} + \frac{x ^{4}}{4} + C

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