integral from 2 to 4 of 1/27 (x^2-2x+3)

解答

\int_{2}^{4} \frac{1}{27} (x^{2} - 2 x + 3) d x

\frac{38}{81}

十进制

0.46913 \dots

求解步骤

\int_{2}^{4} \frac{1}{27} (x^{2} - 2 x + 3) d x

提出常数:

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

= \frac{1}{27} \cdot \int_{2}^{4} x^{2} - 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

= \frac{1}{27} (\int_{2}^{4} x^{2} d x - \int_{2}^{4} 2 x d x + \int_{2}^{4} 3 d x)

\int_{2}^{4} x^{2} d x = \frac{56}{3}

\int_{2}^{4} 2 x d x = 12

\int_{2}^{4} 3 d x = 6

= \frac{1}{27} (\frac{56}{3} - 12 + 6)

化简

\frac{1}{27} (\frac{56}{3} - 12 + 6) : \frac{38}{81}

= \frac{38}{81}

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