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