integral from 0 to 1 of (x-1)^2

解答

\int_{0}^{1} (x - 1)^{2} d x

\frac{1}{3}

十进制

0.33333 \dots

求解步骤

\int_{0}^{1} (x - 1)^{2} d x

乘开

(x - 1)^{2} : x^{2} - 2 x + 1

= \int_{0}^{1} x^{2} - 2 x + 1 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}^{1} x^{2} d x - \int_{0}^{1} 2 x d x + \int_{0}^{1} 1 d x

\int_{0}^{1} x^{2} d x = \frac{1}{3}

\int_{0}^{1} 2 x d x = 1

\int_{0}^{1} 1 d x = 1

= \frac{1}{3} - 1 + 1

化简

= \frac{1}{3}

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