📝 题目
例 3 试计算积分
$$ {\int }_{0}^{\pi /2}{\left( \tan x\right) }^{a}\mathrm{\;d}x\;\left( {\left| \alpha \right| < 1}\right) . $$
💡 答案与解析
解 我们有
$$ {\int }_{0}^{\pi /2}{\left( \tan x\right) }^{a}\mathrm{\;d}x = {\int }_{0}^{\pi /2}{\sin }^{a}x \cdot {\cos }^{-a}x\mathrm{\;d}x $$
$$ = \frac{1}{2}\Gamma \left( \frac{1 + \alpha }{2}\right) \Gamma \left( \frac{1 - \alpha }{2}\right) $$
$$ = \frac{1}{2}\Gamma \left( \frac{1 + \alpha }{2}\right) \Gamma \left( {1 - \frac{1 + \alpha }{2}}\right) $$
$$ = \frac{1}{2}\frac{\pi }{\sin \frac{1 + \alpha }{2}\pi } = \frac{\pi }{2\cos \frac{\alpha \pi }{2}}. $$