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受欢迎的微积分 >

integral of 2xsqrt(4x-5)

解答

\int 2 x 4 x - 5 d x

解答

\frac{1}{30} (6 x + 5) (4 x - 5)^{\frac{3}{2}} + C

求解步骤

\int 2 x 4 x - 5 d x

提出常数:

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

= 2 \cdot \int x 4 x - 5 d x

使用换元积分法

= 2 \cdot \int \frac{u ( u + 5 )}{16} d u

提出常数:

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

= 2 \cdot \frac{1}{16} \cdot \int u (u + 5) d u

乘开

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

= 2 \cdot \frac{1}{16} \cdot \int u^{\frac{3}{2}} + 5 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

= 2 \cdot \frac{1}{16} (\int u^{\frac{3}{2}} d u + \int 5 u d u)

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

\int 5 u d u = \frac{10}{3} u^{\frac{3}{2}}

= 2 \cdot \frac{1}{16} (\frac{2}{5} u^{\frac{5}{2}} + \frac{10}{3} u^{\frac{3}{2}})

u = 4 x - 5

代回

= 2 \cdot \frac{1}{16} (\frac{2}{5} (4 x - 5)^{\frac{5}{2}} + \frac{10}{3} (4 x - 5)^{\frac{3}{2}})

化简

2 \cdot \frac{1}{16} (\frac{2}{5} (4 x - 5)^{\frac{5}{2}} + \frac{10}{3} (4 x - 5)^{\frac{3}{2}}) : \frac{1}{30} (6 x + 5) (4 x - 5)^{\frac{3}{2}}

= \frac{1}{30} (6 x + 5) (4 x - 5)^{\frac{3}{2}}

解答补常数

= \frac{1}{30} (6 x + 5) (4 x - 5)^{\frac{3}{2}} + C

作图

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