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受欢迎的微积分 >

integral of xsqrt(2x-3)

解答

\int x 2 x - 3 d x

解答

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

求解步骤

\int x 2 x - 3 d x

使用换元积分法

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

乘开

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

= \frac{1}{4} \cdot \int u^{\frac{3}{2}} + 3 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 3 u d u)

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

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

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

u = 2 x - 3

代回

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

化简

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

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

解答补常数

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

作图

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