integral of x^5sqrt(x-4)

解答

\int x^{5} x - 4 d x

\frac{2}{13} (x - 4)^{\frac{13}{2}} + \frac{40}{11} (x - 4)^{\frac{11}{2}} + \frac{320}{9} (x - 4)^{\frac{9}{2}} + \frac{1280}{7} (x - 4)^{\frac{7}{2}} + 512 (x - 4)^{\frac{5}{2}} + \frac{2048}{3} (x - 4)^{\frac{3}{2}} + C

求解步骤

\int x^{5} x - 4 d x

使用换元积分法

= \int (u + 4)^{5} u d u

乘开

(u + 4)^{5} u : u^{\frac{11}{2}} + 20 u^{\frac{9}{2}} + 160 u^{\frac{7}{2}} + 640 u^{\frac{5}{2}} + 1280 u^{\frac{3}{2}} + 1024 u

= \int u^{\frac{11}{2}} + 20 u^{\frac{9}{2}} + 160 u^{\frac{7}{2}} + 640 u^{\frac{5}{2}} + 1280 u^{\frac{3}{2}} + 1024 u d u

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 u^{\frac{11}{2}} d u + \int 20 u^{\frac{9}{2}} d u + \int 160 u^{\frac{7}{2}} d u + \int 640 u^{\frac{5}{2}} d u + \int 1280 u^{\frac{3}{2}} d u + \int 1024 u d u

\int u^{\frac{11}{2}} d u = \frac{2}{13} u^{\frac{13}{2}}

\int 20 u^{\frac{9}{2}} d u = \frac{40}{11} u^{\frac{11}{2}}

\int 160 u^{\frac{7}{2}} d u = \frac{320}{9} u^{\frac{9}{2}}

\int 640 u^{\frac{5}{2}} d u = \frac{1280}{7} u^{\frac{7}{2}}

\int 1280 u^{\frac{3}{2}} d u = 512 u^{\frac{5}{2}}

\int 1024 u d u = \frac{2048}{3} u^{\frac{3}{2}}

= \frac{2}{13} u^{\frac{13}{2}} + \frac{40}{11} u^{\frac{11}{2}} + \frac{320}{9} u^{\frac{9}{2}} + \frac{1280}{7} u^{\frac{7}{2}} + 512 u^{\frac{5}{2}} + \frac{2048}{3} u^{\frac{3}{2}}

u = x - 4

代回

= \frac{2}{13} (x - 4)^{\frac{13}{2}} + \frac{40}{11} (x - 4)^{\frac{11}{2}} + \frac{320}{9} (x - 4)^{\frac{9}{2}} + \frac{1280}{7} (x - 4)^{\frac{7}{2}} + 512 (x - 4)^{\frac{5}{2}} + \frac{2048}{3} (x - 4)^{\frac{3}{2}}

解答补常数

= \frac{2}{13} (x - 4)^{\frac{13}{2}} + \frac{40}{11} (x - 4)^{\frac{11}{2}} + \frac{320}{9} (x - 4)^{\frac{9}{2}} + \frac{1280}{7} (x - 4)^{\frac{7}{2}} + 512 (x - 4)^{\frac{5}{2}} + \frac{2048}{3} (x - 4)^{\frac{3}{2}} + C

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