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受欢迎的微积分 >

integral of x^2sqrt(6x+3)

解答

\int x^{2} 6 x + 3 d x

解答

\frac{1}{35 3} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}} + C

求解步骤

\int x^{2} 6 x + 3 d x

使用换元积分法

= \int \frac{u ( u - 3 ) ^{2}}{216} d u

提出常数:

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

= \frac{1}{216} \cdot \int u (u - 3)^{2} d u

乘开

u (u - 3)^{2} : u^{\frac{5}{2}} - 6 u^{\frac{3}{2}} + 9 u

= \frac{1}{216} \cdot \int u^{\frac{5}{2}} - 6 u^{\frac{3}{2}} + 9 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}{216} (\int u^{\frac{5}{2}} d u - \int 6 u^{\frac{3}{2}} d u + \int 9 u d u)

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

\int 6 u^{\frac{3}{2}} d u = \frac{12}{5} u^{\frac{5}{2}}

\int 9 u d u = 6 u^{\frac{3}{2}}

= \frac{1}{216} (\frac{2}{7} u^{\frac{7}{2}} - \frac{12}{5} u^{\frac{5}{2}} + 6 u^{\frac{3}{2}})

u = 6 x + 3

代回

= \frac{1}{216} (\frac{2}{7} (6 x + 3)^{\frac{7}{2}} - \frac{12}{5} (6 x + 3)^{\frac{5}{2}} + 6 (6 x + 3)^{\frac{3}{2}})

化简

\frac{1}{216} (\frac{2}{7} (6 x + 3)^{\frac{7}{2}} - \frac{12}{5} (6 x + 3)^{\frac{5}{2}} + 6 (6 x + 3)^{\frac{3}{2}}) : \frac{1}{35 3} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}}

= \frac{1}{35 3} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}}

解答补常数

= \frac{1}{35 3} (15 x^{2} - 6 x + 2) (2 x + 1)^{\frac{3}{2}} + C

作图

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