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

Lösung

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

\frac{x ^{9}}{9} + \frac{x ^{6}}{3} + \frac{x ^{3}}{3} + C

Schritte zur Lösung

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

Multipliziere aus

x^{2} (x^{3} + 1)^{2} : x^{8} + 2 x^{5} + x^{2}

= \int x^{8} + 2 x^{5} + 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^{8} d x + \int 2 x^{5} d x + \int x^{2} d x

\int x^{8} d x = \frac{x ^{9}}{9}

\int 2 x^{5} d x = \frac{x ^{6}}{3}

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

= \frac{x ^{9}}{9} + \frac{x ^{6}}{3} + \frac{x ^{3}}{3}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{9}}{9} + \frac{x ^{6}}{3} + \frac{x ^{3}}{3} + C

g (x) = \frac{5}{6 x ^{3}}

s im pl i f y x^{2 - 3 x} x^{2}

\int_{1}^{3} \frac{x ^{2} + 6}{4 x - x ^{2}} d x

\int \frac{4 + t + t ^{2}}{t} d t

t an g e n t y = \frac{1}{x - 2}, at x = 6