integral from x^3 to x^5 of (4t+4)^6

解答

\int_{x^{3}}^{x^{5}} (4 t + 4)^{6} d t

4096 x^{30} + 12288 x^{25} + 20480 x^{20} - 4096 x^{18} + 8192 x^{15} - 20480 x^{12} + 12288 x^{10} - 20480 x^{9} - 12288 x^{6} + 4096 x^{5} - 4096 x^{3} + \frac{4096 ( x ^{35} - x ^{21} )}{7}

求解步骤

\int_{x^{3}}^{x^{5}} (4 t + 4)^{6} d t

乘开

(4 t + 4)^{6} : 4096 t^{6} + 24576 t^{5} + 61440 t^{4} + 81920 t^{3} + 61440 t^{2} + 24576 t + 4096

= \int_{x^{3}}^{x^{5}} 4096 t^{6} + 24576 t^{5} + 61440 t^{4} + 81920 t^{3} + 61440 t^{2} + 24576 t + 4096 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_{x^{3}}^{x^{5}} 4096 t^{6} d t + \int_{x^{3}}^{x^{5}} 24576 t^{5} d t + \int_{x^{3}}^{x^{5}} 61440 t^{4} d t + \int_{x^{3}}^{x^{5}} 81920 t^{3} d t + \int_{x^{3}}^{x^{5}} 61440 t^{2} d t + \int_{x^{3}}^{x^{5}} 24576 t d t + \int_{x^{3}}^{x^{5}} 4096 d t

\int_{x^{3}}^{x^{5}} 4096 t^{6} d t = \frac{4096 ( x ^{35} - x ^{21} )}{7}

\int_{x^{3}}^{x^{5}} 24576 t^{5} d t = 4096 (x^{30} - x^{18})

\int_{x^{3}}^{x^{5}} 61440 t^{4} d t = 12288 (x^{25} - x^{15})

\int_{x^{3}}^{x^{5}} 81920 t^{3} d t = 20480 (x^{20} - x^{12})

\int_{x^{3}}^{x^{5}} 61440 t^{2} d t = 20480 (x^{15} - x^{9})

\int_{x^{3}}^{x^{5}} 24576 t d t = 12288 (x^{10} - x^{6})

\int_{x^{3}}^{x^{5}} 4096 d t = 4096 x^{5} - 4096 x^{3}

= \frac{4096 ( x ^{35} - x ^{21} )}{7} + 4096 (x^{30} - x^{18}) + 12288 (x^{25} - x^{15}) + 20480 (x^{20} - x^{12}) + 20480 (x^{15} - x^{9}) + 12288 (x^{10} - x^{6}) + 4096 x^{5} - 4096 x^{3}

化简

\frac{4096 ( x ^{35} - x ^{21} )}{7} + 4096 (x^{30} - x^{18}) + 12288 (x^{25} - x^{15}) + 20480 (x^{20} - x^{12}) + 20480 (x^{15} - x^{9}) + 12288 (x^{10} - x^{6}) + 4096 x^{5} - 4096 x^{3} : 4096 x^{30} + 12288 x^{25} + 20480 x^{20} - 4096 x^{18} + 8192 x^{15} - 20480 x^{12} + 12288 x^{10} - 20480 x^{9} - 12288 x^{6} + 4096 x^{5} - 4096 x^{3} + \frac{4096 ( x ^{35} - x ^{21} )}{7}

= 4096 x^{30} + 12288 x^{25} + 20480 x^{20} - 4096 x^{18} + 8192 x^{15} - 20480 x^{12} + 12288 x^{10} - 20480 x^{9} - 12288 x^{6} + 4096 x^{5} - 4096 x^{3} + \frac{4096 ( x ^{35} - x ^{21} )}{7}

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