integral of x^2(x+1)^3

Lösung

\int x^{2} (x + 1)^{3} d x

\frac{x ^{6}}{6} + \frac{3 x ^{5}}{5} + \frac{3 x ^{4}}{4} + \frac{x ^{3}}{3} + C

Schritte zur Lösung

\int x^{2} (x + 1)^{3} d x

Multipliziere aus

x^{2} (x + 1)^{3} : x^{5} + 3 x^{4} + 3 x^{3} + x^{2}

= \int x^{5} + 3 x^{4} + 3 x^{3} + x^{2} 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^{5} d x + \int 3 x^{4} d x + \int 3 x^{3} d x + \int x^{2} d x

\int x^{5} d x = \frac{x ^{6}}{6}

\int 3 x^{4} d x = \frac{3 x ^{5}}{5}

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

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

= \frac{x ^{6}}{6} + \frac{3 x ^{5}}{5} + \frac{3 x ^{4}}{4} + \frac{x ^{3}}{3}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{6}}{6} + \frac{3 x ^{5}}{5} + \frac{3 x ^{4}}{4} + \frac{x ^{3}}{3} + C

(3 y^{2} + 2 x y) - (2 x y + x^{2}) \frac{d y}{d x} = 0

d er i v a t i v e 3 (s^{3} - s^{2})

d er i v a t i v e f (r) = \frac{6}{r ^{3}}

x \to 2 π lim (x \cdot csc (x))

a re a x^{2} - 12, x - 6