integral from 2 to x of (1/2 t^2-1)^8

解答

\int_{2}^{x} (\frac{1}{2} t^{2} - 1)^{8} d t

- x^{7} + x + \frac{7 ( x ^{5} - 32 )}{5} + \frac{35 ( x ^{9} - 512 )}{72} + \frac{x ^{17} - 131072}{4352} + \frac{7 ( x ^{13} - 8192 )}{208} - \frac{4 ( x ^{3} - 8 )}{3} - \frac{x ^{15} - 32768}{240} - \frac{7 ( x ^{11} - 2048 )}{44} + 126

求解步骤

\int_{2}^{x} (\frac{1}{2} t^{2} - 1)^{8} d t

乘开

(\frac{1}{2} t^{2} - 1)^{8} : \frac{t ^{16}}{256} - \frac{t ^{14}}{16} + \frac{7 t ^{12}}{16} - \frac{7 t ^{10}}{4} + \frac{35 t ^{8}}{8} - 7 t^{6} + 7 t^{4} - 4 t^{2} + 1

= \int_{2}^{x} \frac{t ^{16}}{256} - \frac{t ^{14}}{16} + \frac{7 t ^{12}}{16} - \frac{7 t ^{10}}{4} + \frac{35 t ^{8}}{8} - 7 t^{6} + 7 t^{4} - 4 t^{2} + 1 d t

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_{2}^{x} \frac{t ^{16}}{256} d t - \int_{2}^{x} \frac{t ^{14}}{16} d t + \int_{2}^{x} \frac{7 t ^{12}}{16} d t - \int_{2}^{x} \frac{7 t ^{10}}{4} d t + \int_{2}^{x} \frac{35 t ^{8}}{8} d t - \int_{2}^{x} 7 t^{6} d t + \int_{2}^{x} 7 t^{4} d t - \int_{2}^{x} 4 t^{2} d t + \int_{2}^{x} 1 d t

\int_{2}^{x} \frac{t ^{16}}{256} d t = \frac{x ^{17} - 131072}{4352}

\int_{2}^{x} \frac{t ^{14}}{16} d t = \frac{x ^{15} - 32768}{240}

\int_{2}^{x} \frac{7 t ^{12}}{16} d t = \frac{7 ( x ^{13} - 8192 )}{208}

\int_{2}^{x} \frac{7 t ^{10}}{4} d t = \frac{7 ( x ^{11} - 2048 )}{44}

\int_{2}^{x} \frac{35 t ^{8}}{8} d t = \frac{35 ( x ^{9} - 512 )}{72}

\int_{2}^{x} 7 t^{6} d t = x^{7} - 128

\int_{2}^{x} 7 t^{4} d t = \frac{7 ( x ^{5} - 32 )}{5}

\int_{2}^{x} 4 t^{2} d t = \frac{4 ( x ^{3} - 8 )}{3}

\int_{2}^{x} 1 d t = x - 2

= \frac{x ^{17} - 131072}{4352} - \frac{x ^{15} - 32768}{240} + \frac{7 ( x ^{13} - 8192 )}{208} - \frac{7 ( x ^{11} - 2048 )}{44} + \frac{35 ( x ^{9} - 512 )}{72} - (x^{7} - 128) + \frac{7 ( x ^{5} - 32 )}{5} - \frac{4 ( x ^{3} - 8 )}{3} + x - 2

化简

= - x^{7} + x + \frac{7 ( x ^{5} - 32 )}{5} + \frac{35 ( x ^{9} - 512 )}{72} + \frac{x ^{17} - 131072}{4352} + \frac{7 ( x ^{13} - 8192 )}{208} - \frac{4 ( x ^{3} - 8 )}{3} - \frac{x ^{15} - 32768}{240} - \frac{7 ( x ^{11} - 2048 )}{44} + 126

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