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受欢迎的微积分 >

integral of x^2*sqrt(1-x)

解答

\int x^{2} \cdot 1 - x d x

解答

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

求解步骤

\int x^{2} 1 - x d x

使用换元积分法

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

提出常数:

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

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

乘开

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

= - \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

= - (\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}}

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

u = 1 - x

代回

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

化简

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

解答补常数

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

作图

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\int \frac{lo g _{e} ( x )}{x} d x

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d er i v a t i v e f (x) = ln (10 x)

泰勒 e^{5x+5}

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x \to - \infty lim (x^{2} - x)