integral from 6 to 10 of 3(x-4)^2

解答

\int_{6}^{10} 3 (x - 4)^{2} d x

208

求解步骤

\int_{6}^{10} 3 (x - 4)^{2} d x

提出常数:

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

= 3 \cdot \int_{6}^{10} (x - 4)^{2} d x

乘开

(x - 4)^{2} : x^{2} - 8 x + 16

= 3 \cdot \int_{6}^{10} x^{2} - 8 x + 16 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

= 3 (\int_{6}^{10} x^{2} d x - \int_{6}^{10} 8 x d x + \int_{6}^{10} 16 d x)

\int_{6}^{10} x^{2} d x = \frac{784}{3}

\int_{6}^{10} 8 x d x = 256

\int_{6}^{10} 16 d x = 64

= 3 (\frac{784}{3} - 256 + 64)

化简

3 (\frac{784}{3} - 256 + 64) : 208

= 208

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