$$$- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}$$$ 的积分

求$$$\int \left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)\, dx$$$。

逐项积分：

$${\color{red}{\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)d x}}} = {\color{red}{\left(\int{\frac{1}{2} d x} - \int{\sin{\left(x \right)} d x} + \int{\cos{\left(x \right)} d x}\right)}}$$

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=\frac{1}{2}$$$：

$$- \int{\sin{\left(x \right)} d x} + \int{\cos{\left(x \right)} d x} + {\color{red}{\int{\frac{1}{2} d x}}} = - \int{\sin{\left(x \right)} d x} + \int{\cos{\left(x \right)} d x} + {\color{red}{\left(\frac{x}{2}\right)}}$$

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

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

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

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

因此，

$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)d x} = \frac{x}{2} + \sin{\left(x \right)} + \cos{\left(x \right)}$$

化简：

$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)d x} = \frac{x}{2} + \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$

加上积分常数：

$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)d x} = \frac{x}{2} + \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}+C$$

$$$\int \left(- \sin{\left(x \right)} + \cos{\left(x \right)} + \frac{1}{2}\right)\, dx = \left(\frac{x}{2} + \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}\right) + C$$$A