integral of-4x(3x^2-1)^7

解答

\int - 4 x (3 x^{2} - 1)^{7} d x

- \frac{2187 x ^{16}}{4} + 1458 x^{14} - 1701 x^{12} + 1134 x^{10} - \frac{945 x ^{8}}{2} + 126 x^{6} - 21 x^{4} + 2 x^{2} + C

求解步骤

\int - 4 x (3 x^{2} - 1)^{7} d x

提出常数:

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

= - 4 \cdot \int x (3 x^{2} - 1)^{7} d x

乘开

x (3 x^{2} - 1)^{7} : 2187 x^{15} - 5103 x^{13} + 5103 x^{11} - 2835 x^{9} + 945 x^{7} - 189 x^{5} + 21 x^{3} - x

= - 4 \cdot \int 2187 x^{15} - 5103 x^{13} + 5103 x^{11} - 2835 x^{9} + 945 x^{7} - 189 x^{5} + 21 x^{3} - x d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 4 (\int 2187 x^{15} d x - \int 5103 x^{13} d x + \int 5103 x^{11} d x - \int 2835 x^{9} d x + \int 945 x^{7} d x - \int 189 x^{5} d x + \int 21 x^{3} d x - \int x d x)

\int 2187 x^{15} d x = \frac{2187 x ^{16}}{16}

\int 5103 x^{13} d x = \frac{729 x ^{14}}{2}

\int 5103 x^{11} d x = \frac{1701 x ^{12}}{4}

\int 2835 x^{9} d x = \frac{567 x ^{10}}{2}

\int 945 x^{7} d x = \frac{945 x ^{8}}{8}

\int 189 x^{5} d x = \frac{63 x ^{6}}{2}

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

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

= - 4 (\frac{2187 x ^{16}}{16} - \frac{729 x ^{14}}{2} + \frac{1701 x ^{12}}{4} - \frac{567 x ^{10}}{2} + \frac{945 x ^{8}}{8} - \frac{63 x ^{6}}{2} + \frac{21 x ^{4}}{4} - \frac{x ^{2}}{2})

化简

- 4 (\frac{2187 x ^{16}}{16} - \frac{729 x ^{14}}{2} + \frac{1701 x ^{12}}{4} - \frac{567 x ^{10}}{2} + \frac{945 x ^{8}}{8} - \frac{63 x ^{6}}{2} + \frac{21 x ^{4}}{4} - \frac{x ^{2}}{2}) : - \frac{2187 x ^{16}}{4} + 1458 x^{14} - 1701 x^{12} + 1134 x^{10} - \frac{945 x ^{8}}{2} + 126 x^{6} - 21 x^{4} + 2 x^{2}

= - \frac{2187 x ^{16}}{4} + 1458 x^{14} - 1701 x^{12} + 1134 x^{10} - \frac{945 x ^{8}}{2} + 126 x^{6} - 21 x^{4} + 2 x^{2}

解答补常数

= - \frac{2187 x ^{16}}{4} + 1458 x^{14} - 1701 x^{12} + 1134 x^{10} - \frac{945 x ^{8}}{2} + 126 x^{6} - 21 x^{4} + 2 x^{2} + C

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