integral from 0 to 1 of y^2*3(1-y)^2

解答

\int_{0}^{1} y^{2} \cdot 3 (1 - y)^{2} d y

\frac{1}{10}

十进制

0.1

求解步骤

\int_{0}^{1} y^{2} \cdot 3 (1 - y)^{2} d y

提出常数:

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

= 3 \cdot \int_{0}^{1} y^{2} (1 - y)^{2} d y

乘开

y^{2} (1 - y)^{2} : y^{2} - 2 y^{3} + y^{4}

= 3 \cdot \int_{0}^{1} y^{2} - 2 y^{3} + y^{4} d y

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_{0}^{1} y^{2} d y - \int_{0}^{1} 2 y^{3} d y + \int_{0}^{1} y^{4} d y)

\int_{0}^{1} y^{2} d y = \frac{1}{3}

\int_{0}^{1} 2 y^{3} d y = \frac{1}{2}

\int_{0}^{1} y^{4} d y = \frac{1}{5}

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

化简

3 (\frac{1}{3} - \frac{1}{2} + \frac{1}{5}) : \frac{1}{10}

= \frac{1}{10}

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