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受欢迎的微积分 >

integral of sqrt(x)(sqrt(1+\sqrt{x)})

解答

\int x (1 + x) d x

解答

2 (\frac{2}{7} (1 + x)^{\frac{7}{2}} - \frac{4}{5} (1 + x)^{\frac{5}{2}} + \frac{2}{3} (1 + x)^{\frac{3}{2}}) + C

求解步骤

\int x 1 + x d x

使用换元积分法

= \int 2 u^{2} 1 + u d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int u^{2} 1 + u d u

使用换元积分法

= 2 \cdot \int (v - 1)^{2} v d v

乘开

(v - 1)^{2} v : v^{\frac{5}{2}} - 2 v^{\frac{3}{2}} + v

= 2 \cdot \int v^{\frac{5}{2}} - 2 v^{\frac{3}{2}} + v d v

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

= 2 (\int v^{\frac{5}{2}} d v - \int 2 v^{\frac{3}{2}} d v + \int v d v)

\int v^{\frac{5}{2}} d v = \frac{2}{7} v^{\frac{7}{2}}

\int 2 v^{\frac{3}{2}} d v = \frac{4}{5} v^{\frac{5}{2}}

\int v d v = \frac{2}{3} v^{\frac{3}{2}}

= 2 (\frac{2}{7} v^{\frac{7}{2}} - \frac{4}{5} v^{\frac{5}{2}} + \frac{2}{3} v^{\frac{3}{2}})

代回

= 2 (\frac{2}{7} (1 + x)^{\frac{7}{2}} - \frac{4}{5} (1 + x)^{\frac{5}{2}} + \frac{2}{3} (1 + x)^{\frac{3}{2}})

解答补常数

= 2 (\frac{2}{7} (1 + x)^{\frac{7}{2}} - \frac{4}{5} (1 + x)^{\frac{5}{2}} + \frac{2}{3} (1 + x)^{\frac{3}{2}}) + C

作图

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