integral of (x^2+4x)^7(4x+8)

解答

\int (x^{2} + 4 x)^{7} (4 x + 8) d x

\frac{x ^{16}}{4} + 8 x^{15} + 112 x^{14} + 896 x^{13} + 4480 x^{12} + 14336 x^{11} + 28672 x^{10} + 32768 x^{9} + 16384 x^{8} + C

求解步骤

\int (x^{2} + 4 x)^{7} (4 x + 8) d x

乘开

(x^{2} + 4 x)^{7} (4 x + 8) : 4 x^{15} + 120 x^{14} + 1568 x^{13} + 11648 x^{12} + 53760 x^{11} + 157696 x^{10} + 286720 x^{9} + 294912 x^{8} + 131072 x^{7}

= \int 4 x^{15} + 120 x^{14} + 1568 x^{13} + 11648 x^{12} + 53760 x^{11} + 157696 x^{10} + 286720 x^{9} + 294912 x^{8} + 131072 x^{7} 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

= \int 4 x^{15} d x + \int 120 x^{14} d x + \int 1568 x^{13} d x + \int 11648 x^{12} d x + \int 53760 x^{11} d x + \int 157696 x^{10} d x + \int 286720 x^{9} d x + \int 294912 x^{8} d x + \int 131072 x^{7} d x

\int 4 x^{15} d x = \frac{x ^{16}}{4}

\int 120 x^{14} d x = 8 x^{15}

\int 1568 x^{13} d x = 112 x^{14}

\int 11648 x^{12} d x = 896 x^{13}

\int 53760 x^{11} d x = 4480 x^{12}

\int 157696 x^{10} d x = 14336 x^{11}

\int 286720 x^{9} d x = 28672 x^{10}

\int 294912 x^{8} d x = 32768 x^{9}

\int 131072 x^{7} d x = 16384 x^{8}

= \frac{x ^{16}}{4} + 8 x^{15} + 112 x^{14} + 896 x^{13} + 4480 x^{12} + 14336 x^{11} + 28672 x^{10} + 32768 x^{9} + 16384 x^{8}

解答补常数

= \frac{x ^{16}}{4} + 8 x^{15} + 112 x^{14} + 896 x^{13} + 4480 x^{12} + 14336 x^{11} + 28672 x^{10} + 32768 x^{9} + 16384 x^{8} + C

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