$$$\frac{\sin^{2}{\left(2 \right)}}{x}$$$の積分
入力内容
$$$\int \frac{\sin^{2}{\left(2 \right)}}{x}\, dx$$$ を求めよ。
解答
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\sin^{2}{\left(2 \right)}$$$ と $$$f{\left(x \right)} = \frac{1}{x}$$$ に対して適用する:
$${\color{red}{\int{\frac{\sin^{2}{\left(2 \right)}}{x} d x}}} = {\color{red}{\sin^{2}{\left(2 \right)} \int{\frac{1}{x} d x}}}$$
$$$\frac{1}{x}$$$ の不定積分は $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$ です:
$$\sin^{2}{\left(2 \right)} {\color{red}{\int{\frac{1}{x} d x}}} = \sin^{2}{\left(2 \right)} {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$
したがって、
$$\int{\frac{\sin^{2}{\left(2 \right)}}{x} d x} = \ln{\left(\left|{x}\right| \right)} \sin^{2}{\left(2 \right)}$$
積分定数を加える:
$$\int{\frac{\sin^{2}{\left(2 \right)}}{x} d x} = \ln{\left(\left|{x}\right| \right)} \sin^{2}{\left(2 \right)}+C$$
解答
$$$\int \frac{\sin^{2}{\left(2 \right)}}{x}\, dx = \ln\left(\left|{x}\right|\right) \sin^{2}{\left(2 \right)} + C$$$A