integral from 2 to x of 1/2 (x-2)

解答

\int_{2}^{x} \frac{1}{2} (x - 2) d x

\frac{1}{4} (x^{2} - 4 x + 4)

求解步骤

\int_{2}^{x} \frac{1}{2} (x - 2) d x

提出常数:

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

= \frac{1}{2} \cdot \int_{2}^{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

= \frac{1}{2} (\int_{2}^{x} x d x - \int_{2}^{x} 2 d x)

\int_{2}^{x} x d x = \frac{x ^{2}}{2} - 2

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

= \frac{1}{2} (\frac{x ^{2}}{2} - 2 - (2 x - 4))

化简

\frac{1}{2} (\frac{x ^{2}}{2} - 2 - (2 x - 4)) : \frac{1}{4} (x^{2} - 4 x + 4)

= \frac{1}{4} (x^{2} - 4 x + 4)

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