integral from 0 to 1 of pi(x-x^{10})

解答

\int_{0}^{1} π (x - x^{10}) d x

π \frac{9}{22}

十进制

1.28519 \dots

求解步骤

\int_{0}^{1} π (x - x^{10}) d x

提出常数:

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

= π \cdot \int_{0}^{1} x - x^{10} 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

= π (\int_{0}^{1} x d x - \int_{0}^{1} x^{10} d x)

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

\int_{0}^{1} x^{10} d x = \frac{1}{11}

= π (\frac{1}{2} - \frac{1}{11})

化简

\frac{1}{2} - \frac{1}{11} : \frac{9}{22}

= π \frac{9}{22}

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