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

Lösung

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

\frac{2048 x ^{12}}{3} - 5120 x^{11} + 16000 x^{10} - \frac{80000 x ^{9}}{3} + 25000 x^{8} - 12500 x^{7} + \frac{15625 x ^{6}}{6} + C

Schritte zur Lösung

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

Multipliziere aus

(4 x^{2} - 5 x)^{5} (8 x - 5) : 8192 x^{11} - 56320 x^{10} + 160000 x^{9} - 240000 x^{8} + 200000 x^{7} - 87500 x^{6} + 15625 x^{5}

= \int 8192 x^{11} - 56320 x^{10} + 160000 x^{9} - 240000 x^{8} + 200000 x^{7} - 87500 x^{6} + 15625 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 8192 x^{11} d x - \int 56320 x^{10} d x + \int 160000 x^{9} d x - \int 240000 x^{8} d x + \int 200000 x^{7} d x - \int 87500 x^{6} d x + \int 15625 x^{5} d x

\int 8192 x^{11} d x = \frac{2048 x ^{12}}{3}

\int 56320 x^{10} d x = 5120 x^{11}

\int 160000 x^{9} d x = 16000 x^{10}

\int 240000 x^{8} d x = \frac{80000 x ^{9}}{3}

\int 200000 x^{7} d x = 25000 x^{8}

\int 87500 x^{6} d x = 12500 x^{7}

\int 15625 x^{5} d x = \frac{15625 x ^{6}}{6}

= \frac{2048 x ^{12}}{3} - 5120 x^{11} + 16000 x^{10} - \frac{80000 x ^{9}}{3} + 25000 x^{8} - 12500 x^{7} + \frac{15625 x ^{6}}{6}

Füge eine Konstante zur Lösung hinzu

= \frac{2048 x ^{12}}{3} - 5120 x^{11} + 16000 x^{10} - \frac{80000 x ^{9}}{3} + 25000 x^{8} - 12500 x^{7} + \frac{15625 x ^{6}}{6} + C

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