integral of (x^5+x^3)^{11}(5x^4+3x^2)

解答

\int (x^{5} + x^{3})^{11} (5 x^{4} + 3 x^{2}) d x

\frac{x ^{60}}{12} + x^{58} + \frac{11 x ^{56}}{2} + \frac{55 x ^{54}}{3} + \frac{165 x ^{52}}{4} + 66 x^{50} + 77 x^{48} + 66 x^{46} + \frac{165 x ^{44}}{4} + \frac{55 x ^{42}}{3} + \frac{11 x ^{40}}{2} + x^{38} + \frac{x ^{36}}{12} + C

求解步骤

\int (x^{5} + x^{3})^{11} (5 x^{4} + 3 x^{2}) d x

乘开

(x^{5} + x^{3})^{11} (5 x^{4} + 3 x^{2}) : 5 x^{59} + 58 x^{57} + 308 x^{55} + 990 x^{53} + 2145 x^{51} + 3300 x^{49} + 3696 x^{47} + 3036 x^{45} + 1815 x^{43} + 770 x^{41} + 220 x^{39} + 38 x^{37} + 3 x^{35}

= \int 5 x^{59} + 58 x^{57} + 308 x^{55} + 990 x^{53} + 2145 x^{51} + 3300 x^{49} + 3696 x^{47} + 3036 x^{45} + 1815 x^{43} + 770 x^{41} + 220 x^{39} + 38 x^{37} + 3 x^{35} 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^{59} d x + \int 58 x^{57} d x + \int 308 x^{55} d x + \int 990 x^{53} d x + \int 2145 x^{51} d x + \int 3300 x^{49} d x + \int 3696 x^{47} d x + \int 3036 x^{45} d x + \int 1815 x^{43} d x + \int 770 x^{41} d x + \int 220 x^{39} d x + \int 38 x^{37} d x + \int 3 x^{35} d x

\int 5 x^{59} d x = \frac{x ^{60}}{12}

\int 58 x^{57} d x = x^{58}

\int 308 x^{55} d x = \frac{11 x ^{56}}{2}

\int 990 x^{53} d x = \frac{55 x ^{54}}{3}

\int 2145 x^{51} d x = \frac{165 x ^{52}}{4}

\int 3300 x^{49} d x = 66 x^{50}

\int 3696 x^{47} d x = 77 x^{48}

\int 3036 x^{45} d x = 66 x^{46}

\int 1815 x^{43} d x = \frac{165 x ^{44}}{4}

\int 770 x^{41} d x = \frac{55 x ^{42}}{3}

\int 220 x^{39} d x = \frac{11 x ^{40}}{2}

\int 38 x^{37} d x = x^{38}

\int 3 x^{35} d x = \frac{x ^{36}}{12}

= \frac{x ^{60}}{12} + x^{58} + \frac{11 x ^{56}}{2} + \frac{55 x ^{54}}{3} + \frac{165 x ^{52}}{4} + 66 x^{50} + 77 x^{48} + 66 x^{46} + \frac{165 x ^{44}}{4} + \frac{55 x ^{42}}{3} + \frac{11 x ^{40}}{2} + x^{38} + \frac{x ^{36}}{12}

解答补常数

= \frac{x ^{60}}{12} + x^{58} + \frac{11 x ^{56}}{2} + \frac{55 x ^{54}}{3} + \frac{165 x ^{52}}{4} + 66 x^{50} + 77 x^{48} + 66 x^{46} + \frac{165 x ^{44}}{4} + \frac{55 x ^{42}}{3} + \frac{11 x ^{40}}{2} + x^{38} + \frac{x ^{36}}{12} + C

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