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Beliebt Rechnen >

integral of 10x(4x^2-1)^9

Lösung

\int 10 x (4 x^{2} - 1)^{9} d x

Lösung

10 (\frac{65536 x ^{20}}{5} - 32768 x^{18} + 36864 x^{16} - 24576 x^{14} + 10752 x^{12} - \frac{16128 x ^{10}}{5} + 672 x^{8} - 96 x^{6} + 9 x^{4} - \frac{x ^{2}}{2}) + C

Schritte zur Lösung

\int 10 x (4 x^{2} - 1)^{9} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int x (4 x^{2} - 1)^{9} d x

Multipliziere aus

x (4 x^{2} - 1)^{9} : 262144 x^{19} - 589824 x^{17} + 589824 x^{15} - 344064 x^{13} + 129024 x^{11} - 32256 x^{9} + 5376 x^{7} - 576 x^{5} + 36 x^{3} - x

= 10 \cdot \int 262144 x^{19} - 589824 x^{17} + 589824 x^{15} - 344064 x^{13} + 129024 x^{11} - 32256 x^{9} + 5376 x^{7} - 576 x^{5} + 36 x^{3} - x 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

= 10 (\int 262144 x^{19} d x - \int 589824 x^{17} d x + \int 589824 x^{15} d x - \int 344064 x^{13} d x + \int 129024 x^{11} d x - \int 32256 x^{9} d x + \int 5376 x^{7} d x - \int 576 x^{5} d x + \int 36 x^{3} d x - \int x d x)

\int 262144 x^{19} d x = \frac{65536 x ^{20}}{5}

\int 589824 x^{17} d x = 32768 x^{18}

\int 589824 x^{15} d x = 36864 x^{16}

\int 344064 x^{13} d x = 24576 x^{14}

\int 129024 x^{11} d x = 10752 x^{12}

\int 32256 x^{9} d x = \frac{16128 x ^{10}}{5}

\int 5376 x^{7} d x = 672 x^{8}

\int 576 x^{5} d x = 96 x^{6}

\int 36 x^{3} d x = 9 x^{4}

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

= 10 (\frac{65536 x ^{20}}{5} - 32768 x^{18} + 36864 x^{16} - 24576 x^{14} + 10752 x^{12} - \frac{16128 x ^{10}}{5} + 672 x^{8} - 96 x^{6} + 9 x^{4} - \frac{x ^{2}}{2})

Füge eine Konstante zur Lösung hinzu

= 10 (\frac{65536 x ^{20}}{5} - 32768 x^{18} + 36864 x^{16} - 24576 x^{14} + 10752 x^{12} - \frac{16128 x ^{10}}{5} + 672 x^{8} - 96 x^{6} + 9 x^{4} - \frac{x ^{2}}{2}) + C

Graph

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