integral from-3 to 3 of arccot(x)

解答

\int_{- 3}^{3} arccot (x) d x

3 arccot (3) + 3 arccot (- 3)

十进制

9.42477 \dots

求解步骤

\int_{- 3}^{3} arccot (x) d x

使用分布积分法

= [x arccot (x) - \int - \frac{x}{x ^{2} + 1} d x]_{- 3}^{3}

\int - \frac{x}{x ^{2} + 1} d x = - \frac{1}{2} ln x^{2} + 1

= [x arccot (x) - (- \frac{1}{2} ln x^{2} + 1)]_{- 3}^{3}

化简

= [x arccot (x) + \frac{1}{2} ln x^{2} + 1]_{- 3}^{3}

带入上下限计算后得:

3 arccot (3) + 3 arccot (- 3)

= 3 arccot (3) + 3 arccot (- 3)

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