integral from 0 to 1/7 of arccos(7x)

解答

\int_{0}^{\frac{1}{7}} arccos (7 x) d x

\frac{1}{7}

十进制

0.14285 \dots

求解步骤

\int_{0}^{\frac{1}{7}} arccos (7 x) d x

使用分布积分法

= [x arccos (7 x) - \int - \frac{7 x}{1 - 49 x ^{2}} d x]_{0}^{\frac{1}{7}}

\int - \frac{7 x}{1 - 49 x ^{2}} d x = \frac{1}{7} 1 - 49 x^{2}

= [x arccos (7 x) - \frac{1}{7} 1 - 49 x^{2}]_{0}^{\frac{1}{7}}

化简

[x arccos (7 x) - \frac{1}{7} 1 - 49 x^{2}]_{0}^{\frac{1}{7}} : [\frac{1}{7} (7 x arccos (7 x) - 1 - 49 x^{2})]_{0}^{\frac{1}{7}}

= [\frac{1}{7} (7 x arccos (7 x) - 1 - 49 x^{2})]_{0}^{\frac{1}{7}}

带入上下限计算后得:

\frac{1}{7}

= \frac{1}{7}

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