integral from 0 to 6 of (x-3)^2

解答

\int_{0}^{6} (x - 3)^{2} d x

18

求解步骤

\int_{0}^{6} (x - 3)^{2} d x

乘开

(x - 3)^{2} : x^{2} - 6 x + 9

= \int_{0}^{6} x^{2} - 6 x + 9 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

= \int_{0}^{6} x^{2} d x - \int_{0}^{6} 6 x d x + \int_{0}^{6} 9 d x

\int_{0}^{6} x^{2} d x = 72

\int_{0}^{6} 6 x d x = 108

\int_{0}^{6} 9 d x = 54

= 72 - 108 + 54

化简

= 18

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