integral of (x^2-4x)^5*(4x-8)

Lösung

\int (x^{2} - 4 x)^{5} \cdot (4 x - 8) d x

\frac{x ^{12}}{3} - 8 x^{11} + 80 x^{10} - \frac{1280 x ^{9}}{3} + 1280 x^{8} - 2048 x^{7} + \frac{4096 x ^{6}}{3} + C

Schritte zur Lösung

\int (x^{2} - 4 x)^{5} (4 x - 8) d x

Multipliziere aus

(x^{2} - 4 x)^{5} (4 x - 8) : 4 x^{11} - 88 x^{10} + 800 x^{9} - 3840 x^{8} + 10240 x^{7} - 14336 x^{6} + 8192 x^{5}

= \int 4 x^{11} - 88 x^{10} + 800 x^{9} - 3840 x^{8} + 10240 x^{7} - 14336 x^{6} + 8192 x^{5} 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 4 x^{11} d x - \int 88 x^{10} d x + \int 800 x^{9} d x - \int 3840 x^{8} d x + \int 10240 x^{7} d x - \int 14336 x^{6} d x + \int 8192 x^{5} d x

\int 4 x^{11} d x = \frac{x ^{12}}{3}

\int 88 x^{10} d x = 8 x^{11}

\int 800 x^{9} d x = 80 x^{10}

\int 3840 x^{8} d x = \frac{1280 x ^{9}}{3}

\int 10240 x^{7} d x = 1280 x^{8}

\int 14336 x^{6} d x = 2048 x^{7}

\int 8192 x^{5} d x = \frac{4096 x ^{6}}{3}

= \frac{x ^{12}}{3} - 8 x^{11} + 80 x^{10} - \frac{1280 x ^{9}}{3} + 1280 x^{8} - 2048 x^{7} + \frac{4096 x ^{6}}{3}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{12}}{3} - 8 x^{11} + 80 x^{10} - \frac{1280 x ^{9}}{3} + 1280 x^{8} - 2048 x^{7} + \frac{4096 x ^{6}}{3} + C

\int π^{2} cos (n x) d x

\int (1 - x)^{n - 1} d x

\int \frac{3 y + 1}{y} d y

\int \frac{3}{4 + x} d x

\int 10 - x^{2} (5 x) d x