integral of x(1-2x^2)^9

Lösung

\int x (1 - 2 x^{2})^{9} d x

\frac{x ^{2}}{2} - \frac{9 x ^{4}}{2} + 24 x^{6} - 84 x^{8} + \frac{1008 x ^{10}}{5} - 336 x^{12} + 384 x^{14} - 288 x^{16} + 128 x^{18} - \frac{128 x ^{20}}{5} + C

Schritte zur Lösung

\int x (1 - 2 x^{2})^{9} d x

Multipliziere aus

x (1 - 2 x^{2})^{9} : x - 18 x^{3} + 144 x^{5} - 672 x^{7} + 2016 x^{9} - 4032 x^{11} + 5376 x^{13} - 4608 x^{15} + 2304 x^{17} - 512 x^{19}

= \int x - 18 x^{3} + 144 x^{5} - 672 x^{7} + 2016 x^{9} - 4032 x^{11} + 5376 x^{13} - 4608 x^{15} + 2304 x^{17} - 512 x^{19} 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 d x - \int 18 x^{3} d x + \int 144 x^{5} d x - \int 672 x^{7} d x + \int 2016 x^{9} d x - \int 4032 x^{11} d x + \int 5376 x^{13} d x - \int 4608 x^{15} d x + \int 2304 x^{17} d x - \int 512 x^{19} d x

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

\int 18 x^{3} d x = \frac{9 x ^{4}}{2}

\int 144 x^{5} d x = 24 x^{6}

\int 672 x^{7} d x = 84 x^{8}

\int 2016 x^{9} d x = \frac{1008 x ^{10}}{5}

\int 4032 x^{11} d x = 336 x^{12}

\int 5376 x^{13} d x = 384 x^{14}

\int 4608 x^{15} d x = 288 x^{16}

\int 2304 x^{17} d x = 128 x^{18}

\int 512 x^{19} d x = \frac{128 x ^{20}}{5}

= \frac{x ^{2}}{2} - \frac{9 x ^{4}}{2} + 24 x^{6} - 84 x^{8} + \frac{1008 x ^{10}}{5} - 336 x^{12} + 384 x^{14} - 288 x^{16} + 128 x^{18} - \frac{128 x ^{20}}{5}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - \frac{9 x ^{4}}{2} + 24 x^{6} - 84 x^{8} + \frac{1008 x ^{10}}{5} - 336 x^{12} + 384 x^{14} - 288 x^{16} + 128 x^{18} - \frac{128 x ^{20}}{5} + C