integral from 5 to x of (1/5 t^2-1)^7

解答

\int_{5}^{x} (\frac{1}{5} t^{2} - 1)^{7} d t

\frac{x ^{15}}{1171875} + \frac{x ^{7} - 78125}{25} + \frac{7 ( x ^{3} - 125 )}{15} + \frac{21 ( x ^{11} - 48828125 )}{34375} + 530 - x - \frac{21 x ^{5}}{125} - \frac{7 ( x ^{9} - 1953125 )}{1125} - \frac{7 ( x ^{13} - 5 ^{13} )}{203125} - \frac{78125}{3}

求解步骤

\int_{5}^{x} (\frac{1}{5} t^{2} - 1)^{7} d t

乘开

(\frac{1}{5} t^{2} - 1)^{7} : \frac{t ^{14}}{78125} - \frac{7 t ^{12}}{15625} + \frac{21 t ^{10}}{3125} - \frac{7 t ^{8}}{125} + \frac{7 t ^{6}}{25} - \frac{21 t ^{4}}{25} + \frac{7 t ^{2}}{5} - 1

= \int_{5}^{x} \frac{t ^{14}}{78125} - \frac{7 t ^{12}}{15625} + \frac{21 t ^{10}}{3125} - \frac{7 t ^{8}}{125} + \frac{7 t ^{6}}{25} - \frac{21 t ^{4}}{25} + \frac{7 t ^{2}}{5} - 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_{5}^{x} \frac{t ^{14}}{78125} d t - \int_{5}^{x} \frac{7 t ^{12}}{15625} d t + \int_{5}^{x} \frac{21 t ^{10}}{3125} d t - \int_{5}^{x} \frac{7 t ^{8}}{125} d t + \int_{5}^{x} \frac{7 t ^{6}}{25} d t - \int_{5}^{x} \frac{21 t ^{4}}{25} d t + \int_{5}^{x} \frac{7 t ^{2}}{5} d t - \int_{5}^{x} 1 d t

\int_{5}^{x} \frac{t ^{14}}{78125} d t = \frac{1}{78125} (\frac{x ^{15}}{15} - \frac{5 ^{14}}{3})

\int_{5}^{x} \frac{7 t ^{12}}{15625} d t = \frac{7 ( x ^{13} - 5 ^{13} )}{203125}

\int_{5}^{x} \frac{21 t ^{10}}{3125} d t = \frac{21 ( x ^{11} - 48828125 )}{34375}

\int_{5}^{x} \frac{7 t ^{8}}{125} d t = \frac{7 ( x ^{9} - 1953125 )}{1125}

\int_{5}^{x} \frac{7 t ^{6}}{25} d t = \frac{x ^{7} - 78125}{25}

\int_{5}^{x} \frac{21 t ^{4}}{25} d t = \frac{21}{25} (\frac{x ^{5}}{5} - 625)

\int_{5}^{x} \frac{7 t ^{2}}{5} d t = \frac{7 ( x ^{3} - 125 )}{15}

\int_{5}^{x} 1 d t = x - 5

= \frac{1}{78125} (\frac{x ^{15}}{15} - \frac{5 ^{14}}{3}) - \frac{7 ( x ^{13} - 5 ^{13} )}{203125} + \frac{21 ( x ^{11} - 48828125 )}{34375} - \frac{7 ( x ^{9} - 1953125 )}{1125} + \frac{x ^{7} - 78125}{25} - \frac{21}{25} (\frac{x ^{5}}{5} - 625) + \frac{7 ( x ^{3} - 125 )}{15} - (x - 5)

化简

\frac{1}{78125} (\frac{x ^{15}}{15} - \frac{5 ^{14}}{3}) - \frac{7 ( x ^{13} - 5 ^{13} )}{203125} + \frac{21 ( x ^{11} - 48828125 )}{34375} - \frac{7 ( x ^{9} - 1953125 )}{1125} + \frac{x ^{7} - 78125}{25} - \frac{21}{25} (\frac{x ^{5}}{5} - 625) + \frac{7 ( x ^{3} - 125 )}{15} - (x - 5) : \frac{x ^{15}}{1171875} + \frac{x ^{7} - 78125}{25} + \frac{7 ( x ^{3} - 125 )}{15} + \frac{21 ( x ^{11} - 48828125 )}{34375} + 530 - x - \frac{21 x ^{5}}{125} - \frac{7 ( x ^{9} - 1953125 )}{1125} - \frac{7 ( x ^{13} - 5 ^{13} )}{203125} - \frac{78125}{3}

= \frac{x ^{15}}{1171875} + \frac{x ^{7} - 78125}{25} + \frac{7 ( x ^{3} - 125 )}{15} + \frac{21 ( x ^{11} - 48828125 )}{34375} + 530 - x - \frac{21 x ^{5}}{125} - \frac{7 ( x ^{9} - 1953125 )}{1125} - \frac{7 ( x ^{13} - 5 ^{13} )}{203125} - \frac{78125}{3}

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