integral from 0 to 15 of 2pix(15x-x^2)

解答

\int_{0}^{15} 2 π x (15 x - x^{2}) d x

\frac{16875 π}{2}

十进制

26507.18801 \dots

求解步骤

\int_{0}^{15} 2 π x (15 x - x^{2}) 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}^{15} x (15 x - x^{2}) d x

乘开

x (15 x - x^{2}) : 15 x^{2} - x^{3}

= 2 π \cdot \int_{0}^{15} 15 x^{2} - x^{3} 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}^{15} 15 x^{2} d x - \int_{0}^{15} x^{3} d x)

\int_{0}^{15} 15 x^{2} d x = 16875

\int_{0}^{15} x^{3} d x = \frac{50625}{4}

= 2 π (16875 - \frac{50625}{4})

化简

2 π (16875 - \frac{50625}{4}) : \frac{16875 π}{2}

= \frac{16875 π}{2}

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