integral of (2-2x)^9

Lösung

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

512 x - 2304 x^{2} + 6144 x^{3} - 10752 x^{4} + \frac{64512 x ^{5}}{5} - 10752 x^{6} + 6144 x^{7} - 2304 x^{8} + 512 x^{9} - \frac{256 x ^{10}}{5} + C

Schritte zur Lösung

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

Multipliziere aus

(2 - 2 x)^{9} : 512 - 4608 x + 18432 x^{2} - 43008 x^{3} + 64512 x^{4} - 64512 x^{5} + 43008 x^{6} - 18432 x^{7} + 4608 x^{8} - 512 x^{9}

= \int 512 - 4608 x + 18432 x^{2} - 43008 x^{3} + 64512 x^{4} - 64512 x^{5} + 43008 x^{6} - 18432 x^{7} + 4608 x^{8} - 512 x^{9} 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 512 d x - \int 4608 x d x + \int 18432 x^{2} d x - \int 43008 x^{3} d x + \int 64512 x^{4} d x - \int 64512 x^{5} d x + \int 43008 x^{6} d x - \int 18432 x^{7} d x + \int 4608 x^{8} d x - \int 512 x^{9} d x

\int 512 d x = 512 x

\int 4608 x d x = 2304 x^{2}

\int 18432 x^{2} d x = 6144 x^{3}

\int 43008 x^{3} d x = 10752 x^{4}

\int 64512 x^{4} d x = \frac{64512 x ^{5}}{5}

\int 64512 x^{5} d x = 10752 x^{6}

\int 43008 x^{6} d x = 6144 x^{7}

\int 18432 x^{7} d x = 2304 x^{8}

\int 4608 x^{8} d x = 512 x^{9}

\int 512 x^{9} d x = \frac{256 x ^{10}}{5}

= 512 x - 2304 x^{2} + 6144 x^{3} - 10752 x^{4} + \frac{64512 x ^{5}}{5} - 10752 x^{6} + 6144 x^{7} - 2304 x^{8} + 512 x^{9} - \frac{256 x ^{10}}{5}

Füge eine Konstante zur Lösung hinzu

= 512 x - 2304 x^{2} + 6144 x^{3} - 10752 x^{4} + \frac{64512 x ^{5}}{5} - 10752 x^{6} + 6144 x^{7} - 2304 x^{8} + 512 x^{9} - \frac{256 x ^{10}}{5} + C

y^{^{'}} - y = 11 t e^{2 t}, y (0) = 1

\int 6 x^{2} sin (x) d x

\int_{0}^{\frac{1}{2}} 2 x d x

in v erse l a pl a ce \frac{4}{s ^{3} + 4 s ^{2} + 8 s}

\frac{d y}{d t} + 0.5 t y - 4 t = 0