📝 题目
例 9 求 $\displaystyle{\int \frac{\mathrm{d}x}{\sqrt{{a}^{2} - {x}^{2}}},\int \frac{\mathrm{d}x}{{a}^{2} + {x}^{2}}}$ ,这里 $a > 0$ .
💡 答案与解析
解 $\displaystyle \int \frac{\mathrm{d}x}{\sqrt{{a}^{2} - {x}^{2}}} = \int \frac{\mathrm{d}\left( \frac{x}{a}\right) }{\sqrt{1 - {\left( \frac{x}{a}\right) }^{2}}} = \arcsin \frac{x}{a} + C$ .
$$ \int \frac{\mathrm{d}x}{{a}^{2} + {x}^{2}} = \frac{1}{a}\int \frac{\mathrm{d}\left( \frac{x}{a}\right) }{1 + {\left( \frac{x}{a}\right) }^{2}} = \frac{1}{a}\arctan \frac{x}{a} + C. $$