integral of (x^5+x^2)^{15}(5x^4+2x)

解答

\int (x^{5} + x^{2})^{15} (5 x^{4} + 2 x) d x

\frac{x ^{80}}{16} + x^{77} + \frac{15 x ^{74}}{2} + 35 x^{71} + \frac{455 x ^{68}}{4} + 273 x^{65} + \frac{1001 x ^{62}}{2} + 715 x^{59} + \frac{6435 x ^{56}}{8} + 715 x^{53} + \frac{1001 x ^{50}}{2} + 273 x^{47} + \frac{455 x ^{44}}{4} + 35 x^{41} + \frac{15 x ^{38}}{2} + x^{35} + \frac{x ^{32}}{16} + C

求解步骤

\int (x^{5} + x^{2})^{15} (5 x^{4} + 2 x) d x

乘开

(x^{5} + x^{2})^{15} (5 x^{4} + 2 x) : 5 x^{79} + 77 x^{76} + 555 x^{73} + 2485 x^{70} + 7735 x^{67} + 17745 x^{64} + 31031 x^{61} + 42185 x^{58} + 45045 x^{55} + 37895 x^{52} + 25025 x^{49} + 12831 x^{46} + 5005 x^{43} + 1435 x^{40} + 285 x^{37} + 35 x^{34} + 2 x^{31}

= \int 5 x^{79} + 77 x^{76} + 555 x^{73} + 2485 x^{70} + 7735 x^{67} + 17745 x^{64} + 31031 x^{61} + 42185 x^{58} + 45045 x^{55} + 37895 x^{52} + 25025 x^{49} + 12831 x^{46} + 5005 x^{43} + 1435 x^{40} + 285 x^{37} + 35 x^{34} + 2 x^{31} 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 5 x^{79} d x + \int 77 x^{76} d x + \int 555 x^{73} d x + \int 2485 x^{70} d x + \int 7735 x^{67} d x + \int 17745 x^{64} d x + \int 31031 x^{61} d x + \int 42185 x^{58} d x + \int 45045 x^{55} d x + \int 37895 x^{52} d x + \int 25025 x^{49} d x + \int 12831 x^{46} d x + \int 5005 x^{43} d x + \int 1435 x^{40} d x + \int 285 x^{37} d x + \int 35 x^{34} d x + \int 2 x^{31} d x

\int 5 x^{79} d x = \frac{x ^{80}}{16}

\int 77 x^{76} d x = x^{77}

\int 555 x^{73} d x = \frac{15 x ^{74}}{2}

\int 2485 x^{70} d x = 35 x^{71}

\int 7735 x^{67} d x = \frac{455 x ^{68}}{4}

\int 17745 x^{64} d x = 273 x^{65}

\int 31031 x^{61} d x = \frac{1001 x ^{62}}{2}

\int 42185 x^{58} d x = 715 x^{59}

\int 45045 x^{55} d x = \frac{6435 x ^{56}}{8}

\int 37895 x^{52} d x = 715 x^{53}

\int 25025 x^{49} d x = \frac{1001 x ^{50}}{2}

\int 12831 x^{46} d x = 273 x^{47}

\int 5005 x^{43} d x = \frac{455 x ^{44}}{4}

\int 1435 x^{40} d x = 35 x^{41}

\int 285 x^{37} d x = \frac{15 x ^{38}}{2}

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

\int 2 x^{31} d x = \frac{x ^{32}}{16}

= \frac{x ^{80}}{16} + x^{77} + \frac{15 x ^{74}}{2} + 35 x^{71} + \frac{455 x ^{68}}{4} + 273 x^{65} + \frac{1001 x ^{62}}{2} + 715 x^{59} + \frac{6435 x ^{56}}{8} + 715 x^{53} + \frac{1001 x ^{50}}{2} + 273 x^{47} + \frac{455 x ^{44}}{4} + 35 x^{41} + \frac{15 x ^{38}}{2} + x^{35} + \frac{x ^{32}}{16}

解答补常数

= \frac{x ^{80}}{16} + x^{77} + \frac{15 x ^{74}}{2} + 35 x^{71} + \frac{455 x ^{68}}{4} + 273 x^{65} + \frac{1001 x ^{62}}{2} + 715 x^{59} + \frac{6435 x ^{56}}{8} + 715 x^{53} + \frac{1001 x ^{50}}{2} + 273 x^{47} + \frac{455 x ^{44}}{4} + 35 x^{41} + \frac{15 x ^{38}}{2} + x^{35} + \frac{x ^{32}}{16} + C

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