$$$- \sin{\left(x \right)} + \cos{\left(x \right)}$$$ 的积分
您的输入
求$$$\int \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx$$$。
解答
逐项积分：
$${\color{red}{\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{\sin{\left(x \right)} d x} + \int{\cos{\left(x \right)} d x}\right)}}$$
正弦函数的积分为 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:
$$\int{\cos{\left(x \right)} d x} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = \int{\cos{\left(x \right)} d x} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
余弦函数的积分为 $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$：
$$\cos{\left(x \right)} + {\color{red}{\int{\cos{\left(x \right)} d x}}} = \cos{\left(x \right)} + {\color{red}{\sin{\left(x \right)}}}$$
因此，
$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x} = \sin{\left(x \right)} + \cos{\left(x \right)}$$
化简：
$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x} = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$
加上积分常数：
$$\int{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)d x} = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}+C$$
答案
$$$\int \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} + C$$$A