integral of (x^5+x^3)^{10}(5x^4+3x^2)

解答

\int (x^{5} + x^{3})^{10} (5 x^{4} + 3 x^{2}) d x

\frac{x ^{55}}{11} + x^{53} + 5 x^{51} + 15 x^{49} + 30 x^{47} + 42 x^{45} + 42 x^{43} + 30 x^{41} + 15 x^{39} + 5 x^{37} + x^{35} + \frac{x ^{33}}{11} + C

求解步骤

\int (x^{5} + x^{3})^{10} (5 x^{4} + 3 x^{2}) d x

乘开

(x^{5} + x^{3})^{10} (5 x^{4} + 3 x^{2}) : 5 x^{54} + 53 x^{52} + 255 x^{50} + 735 x^{48} + 1410 x^{46} + 1890 x^{44} + 1806 x^{42} + 1230 x^{40} + 585 x^{38} + 185 x^{36} + 35 x^{34} + 3 x^{32}

= \int 5 x^{54} + 53 x^{52} + 255 x^{50} + 735 x^{48} + 1410 x^{46} + 1890 x^{44} + 1806 x^{42} + 1230 x^{40} + 585 x^{38} + 185 x^{36} + 35 x^{34} + 3 x^{32} 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^{54} d x + \int 53 x^{52} d x + \int 255 x^{50} d x + \int 735 x^{48} d x + \int 1410 x^{46} d x + \int 1890 x^{44} d x + \int 1806 x^{42} d x + \int 1230 x^{40} d x + \int 585 x^{38} d x + \int 185 x^{36} d x + \int 35 x^{34} d x + \int 3 x^{32} d x

\int 5 x^{54} d x = \frac{x ^{55}}{11}

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

\int 255 x^{50} d x = 5 x^{51}

\int 735 x^{48} d x = 15 x^{49}

\int 1410 x^{46} d x = 30 x^{47}

\int 1890 x^{44} d x = 42 x^{45}

\int 1806 x^{42} d x = 42 x^{43}

\int 1230 x^{40} d x = 30 x^{41}

\int 585 x^{38} d x = 15 x^{39}

\int 185 x^{36} d x = 5 x^{37}

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

\int 3 x^{32} d x = \frac{x ^{33}}{11}

= \frac{x ^{55}}{11} + x^{53} + 5 x^{51} + 15 x^{49} + 30 x^{47} + 42 x^{45} + 42 x^{43} + 30 x^{41} + 15 x^{39} + 5 x^{37} + x^{35} + \frac{x ^{33}}{11}

解答补常数

= \frac{x ^{55}}{11} + x^{53} + 5 x^{51} + 15 x^{49} + 30 x^{47} + 42 x^{45} + 42 x^{43} + 30 x^{41} + 15 x^{39} + 5 x^{37} + x^{35} + \frac{x ^{33}}{11} + C

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