integral from 0 to pi of cot(x)tan(x)

解答

\int_{0}^{π} cot (x) tan (x) d x

π

十进制

3.14159 \dots

求解步骤

\int_{0}^{π} cot (x) tan (x) d x

取

cot (x) tan (x)

的分母，令其等于零:

x = \frac{π}{2}

如果存在

b, a < b < c, f (b) =

无定义

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{0}^{\frac{π}{2}} cot (x) tan (x) d x + \int_{\frac{π}{2}}^{π} cot (x) tan (x) d x

\int_{0}^{\frac{π}{2}} cot (x) tan (x) d x = \frac{π}{2}

\int_{\frac{π}{2}}^{π} cot (x) tan (x) d x = π - \frac{π}{2}

= \frac{π}{2} + π - \frac{π}{2}

化简

= π

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