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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

x^{2} 3 x + 3 = 3 x^{2} x + 1

= \int 3 x^{2} x + 1 d x

提出常数:

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

= 3 \cdot \int x^{2} x + 1 d x

使用换元积分法

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

乘开

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

= 3 \cdot \int u^{\frac{5}{2}} - 2 u^{\frac{3}{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

= 3 (\int u^{\frac{5}{2}} d u - \int 2 u^{\frac{3}{2}} d u + \int u d u)

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

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

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

= 3 (\frac{2}{7} u^{\frac{7}{2}} - \frac{4}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}})

u = x + 1

代回

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

解答补常数

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

作图

流行的例子

derivative of 4x^2-x

\frac{d}{d x} (4 x^{2} - x)

integral of 4x(2x^2+1)^8

\int 4 x (2 x^{2} + 1)^{8} d x

tangent f(x)=x^2-5,\at x=1

t an g e n t f (x) = x^{2} - 5, at x = 1

泰勒 x^3,\at 16/3

t a y l or x^{3}, at \frac{16}{3}

y^{''}-4y^'-5y=0,y(1)=0,y^'(1)=8

y^{^{''}} - 4 y^{^{'}} - 5 y = 0, y (1) = 0, y^{^{'}} (1) = 8