$$$- \frac{a^{2}}{\sin{\left(x \right)}} + \sin{\left(x \right)}$$$ 关于$$$x$$$的积分

求$$$\int \left(- \frac{a^{2}}{\sin{\left(x \right)}} + \sin{\left(x \right)}\right)\, dx$$$。

逐项积分：

$${\color{red}{\int{\left(- \frac{a^{2}}{\sin{\left(x \right)}} + \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{\frac{a^{2}}{\sin{\left(x \right)}} d x} + \int{\sin{\left(x \right)} d x}\right)}}$$

使用二倍角公式 $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$ 改写正弦:

$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{\frac{a^{2}}{\sin{\left(x \right)}} d x}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{\int{\frac{a^{2}}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} d x}}}$$

将分子和分母同时乘以 $$$\sec^2\left(\frac{x}{2} \right)$$$:

$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{\frac{a^{2}}{2 \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}} d x}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{\int{\frac{a^{2} \sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} d x}}}$$

设$$$u=\tan{\left(\frac{x}{2} \right)}$$$。

则$$$du=\left(\tan{\left(\frac{x}{2} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2} dx$$$ (步骤见»)，并有$$$\sec^{2}{\left(\frac{x}{2} \right)} dx = 2 du$$$。

该积分可以改写为

$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{\frac{a^{2} \sec^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} d x}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{\int{\frac{a^{2}}{u} d u}}}$$

对 $$$c=a^{2}$$$ 和 $$$f{\left(u \right)} = \frac{1}{u}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{\frac{a^{2}}{u} d u}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{a^{2} \int{\frac{1}{u} d u}}}$$

$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$- a^{2} {\color{red}{\int{\frac{1}{u} d u}}} + \int{\sin{\left(x \right)} d x} = - a^{2} {\color{red}{\ln{\left(\left|{u}\right| \right)}}} + \int{\sin{\left(x \right)} d x}$$

回忆一下 $$$u=\tan{\left(\frac{x}{2} \right)}$$$:

$$- a^{2} \ln{\left(\left|{{\color{red}{u}}}\right| \right)} + \int{\sin{\left(x \right)} d x} = - a^{2} \ln{\left(\left|{{\color{red}{\tan{\left(\frac{x}{2} \right)}}}}\right| \right)} + \int{\sin{\left(x \right)} d x}$$

正弦函数的积分为 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$- a^{2} \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} + {\color{red}{\int{\sin{\left(x \right)} d x}}} = - a^{2} \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

因此，

$$\int{\left(- \frac{a^{2}}{\sin{\left(x \right)}} + \sin{\left(x \right)}\right)d x} = - a^{2} \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} - \cos{\left(x \right)}$$

加上积分常数：

$$\int{\left(- \frac{a^{2}}{\sin{\left(x \right)}} + \sin{\left(x \right)}\right)d x} = - a^{2} \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} - \cos{\left(x \right)}+C$$

$$$\int \left(- \frac{a^{2}}{\sin{\left(x \right)}} + \sin{\left(x \right)}\right)\, dx = \left(- a^{2} \ln\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right|\right) - \cos{\left(x \right)}\right) + C$$$A