integral of xsqrt(x+10)

解答

\int x x + 10 d x

\frac{2}{5} (x + 10)^{\frac{5}{2}} - \frac{20}{3} (x + 10)^{\frac{3}{2}} + C

求解步骤

\int x x + 10 d x

使用换元积分法

= \int (u - 10) u d u

乘开

(u - 10) u : u^{\frac{3}{2}} - 10 u

= \int u^{\frac{3}{2}} - 10 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{3}{2}} d u - \int 10 u d u

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

\int 10 u d u = \frac{20}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} - \frac{20}{3} u^{\frac{3}{2}}

u = x + 10

代回

= \frac{2}{5} (x + 10)^{\frac{5}{2}} - \frac{20}{3} (x + 10)^{\frac{3}{2}}

解答补常数

= \frac{2}{5} (x + 10)^{\frac{5}{2}} - \frac{20}{3} (x + 10)^{\frac{3}{2}} + C