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$$$\sin{\left(x \right)} - \cos{\left(x \right)}$$$の積分
関連する計算機: 定積分・広義積分計算機
入力内容
$$$\int \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)\, dx$$$ を求めよ。
解答
項別に積分せよ:
$${\color{red}{\int{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{\sin{\left(x \right)} d x} - \int{\cos{\left(x \right)} d x}\right)}}$$
余弦の積分は$$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:
$$\int{\sin{\left(x \right)} d x} - {\color{red}{\int{\cos{\left(x \right)} d x}}} = \int{\sin{\left(x \right)} d x} - {\color{red}{\sin{\left(x \right)}}}$$
正弦関数の不定積分は$$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$です:
$$- \sin{\left(x \right)} + {\color{red}{\int{\sin{\left(x \right)} d x}}} = - \sin{\left(x \right)} + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
したがって、
$$\int{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)d x} = - \sin{\left(x \right)} - \cos{\left(x \right)}$$
簡単化せよ:
$$\int{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)d x} = - \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$
積分定数を加える:
$$\int{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)d x} = - \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}+C$$
解答
$$$\int \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)\, dx = - \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} + C$$$A