题目

(int )_(dfrac {3)(4)}^1dfrac (dx)(sqrt {1-x)-1}.

$\int_{\frac34 }^{1 }\frac{dx}{\sqrt{1-x}-1}$ .

题目解答

答案

解：

$\begin{aligned} &令t=\sqrt{1-x},t^{2}=1-x \\ &得到dx=-2tdt,目由\frac{3}{4}\leq x\leq1, \\ &可得0\leq t\leq\frac{1}{2}. \\ &\int_{\frac{3}{4}}^{1}\frac{dx}{\sqrt{1-x}-1} \\ &=\int_{0}^{\frac{1}{2}}\frac{-2tdt}{t-1} \\ &=\int_{0}^{\frac{1}{2}}\frac{-2t+2-2}{t-1}dt \\ &=\int_{0}^{\frac{1}{2}}-2-\frac{2}{t+1}dt \\ &=-2t\vert_{0}^{\frac{1}{2}}-2\int_{0}^{\frac{1}{2}}\frac{1}{t-1}dt \\ &=-1-2\ln|t-1||_{0}^{\frac{1}{2}}+C \\ &=-1-2\ln\frac{1}{2}+C =2\ln2-1+C \end{aligned}$