integral of x(1-7x^2)^6

Lösung

\int x (1 - 7 x^{2})^{6} d x

\frac{x ^{2}}{2} - \frac{21 x ^{4}}{2} + \frac{245 x ^{6}}{2} - \frac{1715 x ^{8}}{2} + \frac{7203 x ^{10}}{2} - \frac{16807 x ^{12}}{2} + \frac{16807 x ^{14}}{2} + C

Schritte zur Lösung

\int x (1 - 7 x^{2})^{6} d x

Multipliziere aus

x (1 - 7 x^{2})^{6} : x - 42 x^{3} + 735 x^{5} - 6860 x^{7} + 36015 x^{9} - 100842 x^{11} + 117649 x^{13}

= \int x - 42 x^{3} + 735 x^{5} - 6860 x^{7} + 36015 x^{9} - 100842 x^{11} + 117649 x^{13} 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 42 x^{3} d x + \int 735 x^{5} d x - \int 6860 x^{7} d x + \int 36015 x^{9} d x - \int 100842 x^{11} d x + \int 117649 x^{13} d x

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

\int 42 x^{3} d x = \frac{21 x ^{4}}{2}

\int 735 x^{5} d x = \frac{245 x ^{6}}{2}

\int 6860 x^{7} d x = \frac{1715 x ^{8}}{2}

\int 36015 x^{9} d x = \frac{7203 x ^{10}}{2}

\int 100842 x^{11} d x = \frac{16807 x ^{12}}{2}

\int 117649 x^{13} d x = \frac{16807 x ^{14}}{2}

= \frac{x ^{2}}{2} - \frac{21 x ^{4}}{2} + \frac{245 x ^{6}}{2} - \frac{1715 x ^{8}}{2} + \frac{7203 x ^{10}}{2} - \frac{16807 x ^{12}}{2} + \frac{16807 x ^{14}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - \frac{21 x ^{4}}{2} + \frac{245 x ^{6}}{2} - \frac{1715 x ^{8}}{2} + \frac{7203 x ^{10}}{2} - \frac{16807 x ^{12}}{2} + \frac{16807 x ^{14}}{2} + C

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