integral from 0 to 1 of 2pix(x-x^3)

解答

\int_{0}^{1} 2 π x (x - x^{3}) d x

\frac{4 π}{15}

十进制

0.83775 \dots

求解步骤

\int_{0}^{1} 2 π x (x - x^{3}) d x

提出常数:

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

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

乘开

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

= 2 π \cdot \int_{0}^{1} x^{2} - 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

= 2 π (\int_{0}^{1} x^{2} d x - \int_{0}^{1} x^{4} d x)

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

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

= 2 π (\frac{1}{3} - \frac{1}{5})

化简

2 π (\frac{1}{3} - \frac{1}{5}) : \frac{4 π}{15}

= \frac{4 π}{15}

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