integral from 4 to 12 of (4-x)(x-12)

解答

\int_{4}^{12} (4 - x) (x - 12) d x

\frac{256}{3}

十进制

85.33333 \dots

求解步骤

\int_{4}^{12} (4 - x) (x - 12) d x

乘开

(4 - x) (x - 12) : 16 x - x^{2} - 48

= \int_{4}^{12} 16 x - x^{2} - 48 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_{4}^{12} 16 x d x - \int_{4}^{12} x^{2} d x - \int_{4}^{12} 48 d x

\int_{4}^{12} 16 x d x = 1024

\int_{4}^{12} x^{2} d x = \frac{1664}{3}

\int_{4}^{12} 48 d x = 384

= 1024 - \frac{1664}{3} - 384

化简

1024 - \frac{1664}{3} - 384 : \frac{256}{3}

= \frac{256}{3}

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