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受欢迎的微积分 >

integral of x^2sqrt(7x-5)

解答

\int x^{2} 7 x - 5 d x

解答

\frac{1}{343} (\frac{2}{7} (7 x - 5)^{\frac{7}{2}} + 4 (7 x - 5)^{\frac{5}{2}} + \frac{50}{3} (7 x - 5)^{\frac{3}{2}}) + C

求解步骤

\int x^{2} 7 x - 5 d x

使用换元积分法

= \int \frac{u ( u + 5 ) ^{2}}{343} d u

提出常数:

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

= \frac{1}{343} \cdot \int u (u + 5)^{2} d u

乘开

u (u + 5)^{2} : u^{\frac{5}{2}} + 10 u^{\frac{3}{2}} + 25 u

= \frac{1}{343} \cdot \int u^{\frac{5}{2}} + 10 u^{\frac{3}{2}} + 25 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}{343} (\int u^{\frac{5}{2}} d u + \int 10 u^{\frac{3}{2}} d u + \int 25 u d u)

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

\int 10 u^{\frac{3}{2}} d u = 4 u^{\frac{5}{2}}

\int 25 u d u = \frac{50}{3} u^{\frac{3}{2}}

= \frac{1}{343} (\frac{2}{7} u^{\frac{7}{2}} + 4 u^{\frac{5}{2}} + \frac{50}{3} u^{\frac{3}{2}})

u = 7 x - 5

代回

= \frac{1}{343} (\frac{2}{7} (7 x - 5)^{\frac{7}{2}} + 4 (7 x - 5)^{\frac{5}{2}} + \frac{50}{3} (7 x - 5)^{\frac{3}{2}})

解答补常数

= \frac{1}{343} (\frac{2}{7} (7 x - 5)^{\frac{7}{2}} + 4 (7 x - 5)^{\frac{5}{2}} + \frac{50}{3} (7 x - 5)^{\frac{3}{2}}) + C

作图

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