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受欢迎的微积分 >

integral of x(1-x)^{1/2}

解答

\int x (1 - x)^{\frac{1}{2}} d x

解答

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

求解步骤

\int x (1 - x)^{\frac{1}{2}} d x

使用换元积分法

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

提出常数:

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

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

乘开

(- u^{2} + 1) u^{2} : - u^{4} + u^{2}

= - 2 \cdot \int - u^{4} + u^{2} 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 (- \int u^{4} d u + \int u^{2} d u)

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

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

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

u = (1 - x)^{\frac{1}{2}}

代回

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

化简

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

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

解答补常数

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

作图

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