integral from pi/4 to pi/2 of (-2cot(x)csc^2(x))

解答

\int_{\frac{π}{4}}^{\frac{π}{2}} (- 2 cot (x) csc^{2} (x)) d x

- 1

求解步骤

\int_{\frac{π}{4}}^{\frac{π}{2}} - 2 cot (x) csc^{2} (x) d x

提出常数:

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

= - 2 \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} cot (x) csc^{2} (x) d x

使用换元积分法

= - 2 \cdot \int_{2}^{1} - \frac{1}{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - 2 (- \int_{1}^{2} - \frac{1}{2} d u)

常数积分:

\int a d x = a x

= - 2 (- [(- \frac{1}{2}) u]_{1}^{2})

化简

= 2 [- \frac{1}{2} u]_{1}^{2}

带入上下限计算后得:

- \frac{1}{2}

= 2 (- \frac{1}{2})

化简

= - 1

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