📝 题目
例 13 求 $\displaystyle{\int \frac{\mathrm{d}x}{\sqrt{{x}^{2} + {a}^{2}}}}$ ,这里 $a > 0$
💡 答案与解析
解 令 $x = a\tan t$ ,则 $\mathrm{d}x = \frac{a\mathrm{\;d}t}{{\cos }^{2}t}$ . 于是
$$ \int \frac{\mathrm{d}x}{\sqrt{{x}^{2} + {a}^{2}}} = \int \frac{\mathrm{d}t}{\cos t} $$
$$ = \ln \left| {\sec t + \tan t}\right| + {C}_{0} $$
$$ = \ln \left| {\frac{\sqrt{{x}^{2} + {a}^{2}}}{a} + \frac{x}{a}}\right| + {C}_{0} $$
$$ = \ln \left| {x + \sqrt{{x}^{2} + {a}^{2}}}\right| + C, $$
这里 $C = {C}_{0} - \ln a$ 仍是任意常数.