integral from 0 to 10 of c(10x-x^2)

解答

\int_{0}^{10} c (10 x - x^{2}) d x

\frac{500}{3} c

求解步骤

\int_{0}^{10} c (10 x - x^{2}) d x

提出常数:

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

= c \cdot \int_{0}^{10} 10 x - x^{2} 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

= c (\int_{0}^{10} 10 x d x - \int_{0}^{10} x^{2} d x)

\int_{0}^{10} 10 x d x = 500

\int_{0}^{10} x^{2} d x = \frac{1000}{3}

= c (500 - \frac{1000}{3})

化简

500 - \frac{1000}{3} : \frac{500}{3}

= \frac{500}{3} c

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