integral from 0 to 1 of 7x^6(6-x^7)

解答

\int_{0}^{1} 7 x^{6} (6 - x^{7}) d x

\frac{11}{2}

十进制

5.5

求解步骤

\int_{0}^{1} 7 x^{6} (6 - x^{7}) d x

提出常数:

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

= 7 \cdot \int_{0}^{1} x^{6} (6 - x^{7}) d x

乘开

x^{6} (6 - x^{7}) : 6 x^{6} - x^{13}

= 7 \cdot \int_{0}^{1} 6 x^{6} - x^{13} 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

= 7 (\int_{0}^{1} 6 x^{6} d x - \int_{0}^{1} x^{13} d x)

\int_{0}^{1} 6 x^{6} d x = \frac{6}{7}

\int_{0}^{1} x^{13} d x = \frac{1}{14}

= 7 (\frac{6}{7} - \frac{1}{14})

化简

7 (\frac{6}{7} - \frac{1}{14}) : \frac{11}{2}

= \frac{11}{2}

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