方企勤 第三章 一元函数积分学 第8题
📝 题目
例 8 设 $a < b$ ,求 $\displaystyle \int \sqrt{\left( {x - a}\right) \left( {b - x}\right) }\mathrm{d}x$ .
💡 答案解析
解法 1 由配方得到
$$
\left( {x - a}\right) \left( {b - x}\right) = {R}^{2} - {\left( x - \frac{a + b}{2}\right) }^{2},
$$
其中 $R\overset{\text{ 定义 }}{ = }\frac{b - a}{2}$ . 作变量代换 $x = u + \frac{a + b}{2}$ ,则有
$$
\text{ 原式 } = \int \sqrt{{R}^{2} - {u}^{2}}\frac{u = R\sin t}{}{R}^{2}\int {\cos }^{2}t\mathrm{\;d}t
$$
$$
= {R}^{2}\int \frac{1 + \cos {2t}}{2}\mathrm{\;d}t = {R}^{2}\left( {\frac{t}{2} + \frac{1}{4}\sin {2t}}\right) + C
$$
$$
= \frac{{R}^{2}}{2}t + \frac{{R}^{2}}{2}\sin t\cos t + C
$$
$$
= \frac{{R}^{2}}{2}\arcsin \frac{u}{R} + \frac{u}{2}\sqrt{{R}^{2} - {u}^{2}} + C
$$
$$
= \frac{1}{4}{\left( b - a\right) }^{2}\arcsin \frac{{2x} - \left( {a + b}\right) }{b - a}
$$
$$
+ \frac{{2x} - \left( {a + b}\right) }{4}\sqrt{\left( {x - a}\right) \left( {b - x}\right) } + C.
$$
解法 2 因为被积函数的定义域为(a, b),所以可设 $x - a =$ $\left( {b - a}\right) {\sin }^{2}t\left( {0 < t < \frac{\pi }{2}}\right)$ . 从而
$$
\sqrt{\left( {x - a}\right) \left( {b - x}\right) } = \left( {b - a}\right) \sin t\cos t,
$$
$$
\mathrm{d}x = 2\left( {b - a}\right) \sin t\cos t\mathrm{\;d}t,
$$
$$
\int \sqrt{\left( {x - a}\right) \left( {b - x}\right) }\mathrm{d}x = 2{\left( b - a\right) }^{2}\int {\sin }^{2}t{\cos }^{2}t\mathrm{\;d}t
$$
$$
= \frac{1}{2}{\left( b - a\right) }^{2}\int {\sin }^{2}{2t}\mathrm{\;d}t = \frac{1}{4}{\left( b - a\right) }^{2}\int \left( {1 - \cos {4t}}\right) \mathrm{d}t
$$
$$
= \frac{1}{4}{\left( b - a\right) }^{2}\left( {t - \sin {4t}}\right) + C. \tag{1.1}
$$
又注意到
$$
\sin {4t} = 4\sin t\cos t\left( {1 - 2{\sin }^{2}t}\right)
$$
$$
= 4\sqrt{\frac{x - a}{b - a}} \cdot \sqrt{1 - \frac{x - a}{b - a}}\left( {1 - 2 \cdot \frac{x - a}{b - a}}\right)
$$
$$
= - 4\frac{{2x} - \left( {a + b}\right) }{{\left( b - a\right) }^{2}}\sqrt{\left( {x - a}\right) \left( {b - x}\right) },
$$
故有
$$
\int \sqrt{\left( {x - a}\right) \left( {b - x}\right) }\mathrm{d}x = \frac{1}{4}{\left( b - a\right) }^{2}\arcsin \sqrt{\frac{x - a}{b - a}}
$$
$$
+ \frac{{2x} - \left( {a + b}\right) }{4}\sqrt{\left( {x - a}\right) \left( {b - x}\right) } + C.
$$
\subsubsection{三、联合求解法}