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

Lösung

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

\frac{x ^{3}}{3} + \frac{x ^{4}}{2} + \frac{x ^{5}}{5} + C

Schritte zur Lösung

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

Multipliziere aus

x^{2} (1 + x)^{2} : x^{2} + 2 x^{3} + x^{4}

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

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

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

\int x^{4} d x = \frac{x ^{5}}{5}

= \frac{x ^{3}}{3} + \frac{x ^{4}}{2} + \frac{x ^{5}}{5}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + \frac{x ^{4}}{2} + \frac{x ^{5}}{5} + C

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