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受欢迎的微积分 >

integral of xsqrt(2+2x)

解答

\int x 2 + 2 x d x

解答

\frac{2 2}{15} (3 x - 2) (x + 1)^{\frac{3}{2}} + C

求解步骤

\int x 2 + 2 x d x

使用换元积分法

= \int \frac{u ( u - 2 )}{4} d u

提出常数:

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

= \frac{1}{4} \cdot \int u (u - 2) d u

乘开

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

= \frac{1}{4} \cdot \int u^{\frac{3}{2}} - 2 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

= \frac{1}{4} (\int u^{\frac{3}{2}} d u - \int 2 u d u)

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

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

= \frac{1}{4} (\frac{2}{5} u^{\frac{5}{2}} - \frac{4}{3} u^{\frac{3}{2}})

u = 2 + 2 x

代回

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

化简

\frac{1}{4} (\frac{2}{5} (2 + 2 x)^{\frac{5}{2}} - \frac{4}{3} (2 + 2 x)^{\frac{3}{2}}) : \frac{2 2}{15} (3 x - 2) (x + 1)^{\frac{3}{2}}

= \frac{2 2}{15} (3 x - 2) (x + 1)^{\frac{3}{2}}

解答补常数

= \frac{2 2}{15} (3 x - 2) (x + 1)^{\frac{3}{2}} + C

作图

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