integral from 0 to 0.4 of 30x^2(1-x)^2

解答

\int_{0}^{0.4} 30 x^{2} (1 - x)^{2} d x

0.31744

求解步骤

\int_{0}^{0.4} 30 x^{2} (1 - x)^{2} d x

提出常数:

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

= 30 \cdot \int_{0}^{0.4} x^{2} (1 - x)^{2} d x

乘开

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

= 30 \cdot \int_{0}^{0.4} x^{2} - 2 x^{3} + x^{4} d x

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

= 30 (\int_{0}^{0.4} x^{2} d x - \int_{0}^{0.4} 2 x^{3} d x + \int_{0}^{0.4} x^{4} d x)

\int_{0}^{0.4} x^{2} d x = 0.02133 \dots

\int_{0}^{0.4} 2 x^{3} d x = 0.0128

\int_{0}^{0.4} x^{4} d x = 0.002048

= 30 (0.02133 \dots - 0.0128 + 0.002048)

化简

= 0.31744

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