integral of x^3sqrt(x-3)

解答

\int x^{3} x - 3 d x

\frac{2}{9} (x - 3)^{\frac{9}{2}} + \frac{18}{7} (x - 3)^{\frac{7}{2}} + \frac{54}{5} (x - 3)^{\frac{5}{2}} + 18 (x - 3)^{\frac{3}{2}} + C

求解步骤

\int x^{3} x - 3 d x

使用换元积分法

= \int (u + 3)^{3} u d u

乘开

(u + 3)^{3} u : u^{\frac{7}{2}} + 9 u^{\frac{5}{2}} + 27 u^{\frac{3}{2}} + 27 u

= \int u^{\frac{7}{2}} + 9 u^{\frac{5}{2}} + 27 u^{\frac{3}{2}} + 27 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{7}{2}} d u + \int 9 u^{\frac{5}{2}} d u + \int 27 u^{\frac{3}{2}} d u + \int 27 u d u

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

\int 9 u^{\frac{5}{2}} d u = \frac{18}{7} u^{\frac{7}{2}}

\int 27 u^{\frac{3}{2}} d u = \frac{54}{5} u^{\frac{5}{2}}

\int 27 u d u = 18 u^{\frac{3}{2}}

= \frac{2}{9} u^{\frac{9}{2}} + \frac{18}{7} u^{\frac{7}{2}} + \frac{54}{5} u^{\frac{5}{2}} + 18 u^{\frac{3}{2}}

u = x - 3

代回

= \frac{2}{9} (x - 3)^{\frac{9}{2}} + \frac{18}{7} (x - 3)^{\frac{7}{2}} + \frac{54}{5} (x - 3)^{\frac{5}{2}} + 18 (x - 3)^{\frac{3}{2}}

解答补常数

= \frac{2}{9} (x - 3)^{\frac{9}{2}} + \frac{18}{7} (x - 3)^{\frac{7}{2}} + \frac{54}{5} (x - 3)^{\frac{5}{2}} + 18 (x - 3)^{\frac{3}{2}} + C

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