integral of (7+x^7)^6x^6

解答

\int (7 + x^{7})^{6} x^{6} d x

16807 x^{7} + 7203 x^{14} + 1715 x^{21} + 245 x^{28} + 21 x^{35} + x^{42} + \frac{x ^{49}}{49} + C

求解步骤

\int (7 + x^{7})^{6} x^{6} d x

乘开

(7 + x^{7})^{6} x^{6} : 117649 x^{6} + 100842 x^{13} + 36015 x^{20} + 6860 x^{27} + 735 x^{34} + 42 x^{41} + x^{48}

= \int 117649 x^{6} + 100842 x^{13} + 36015 x^{20} + 6860 x^{27} + 735 x^{34} + 42 x^{41} + x^{48} 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 117649 x^{6} d x + \int 100842 x^{13} d x + \int 36015 x^{20} d x + \int 6860 x^{27} d x + \int 735 x^{34} d x + \int 42 x^{41} d x + \int x^{48} d x

\int 117649 x^{6} d x = 16807 x^{7}

\int 100842 x^{13} d x = 7203 x^{14}

\int 36015 x^{20} d x = 1715 x^{21}

\int 6860 x^{27} d x = 245 x^{28}

\int 735 x^{34} d x = 21 x^{35}

\int 42 x^{41} d x = x^{42}

\int x^{48} d x = \frac{x ^{49}}{49}

= 16807 x^{7} + 7203 x^{14} + 1715 x^{21} + 245 x^{28} + 21 x^{35} + x^{42} + \frac{x ^{49}}{49}

解答补常数

= 16807 x^{7} + 7203 x^{14} + 1715 x^{21} + 245 x^{28} + 21 x^{35} + x^{42} + \frac{x ^{49}}{49} + C

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