$$$\frac{f_{1} \tan^{2}{\left(f \right)}}{g}$$$ の $$$g$$$ に関する積分

$$$\int \frac{f_{1} \tan^{2}{\left(f \right)}}{g}\, dg$$$ を求めよ。

定数倍の法則 $$$\int c f{\left(g \right)}\, dg = c \int f{\left(g \right)}\, dg$$$ を、$$$c=f_{1} \tan^{2}{\left(f \right)}$$$ と $$$f{\left(g \right)} = \frac{1}{g}$$$ に対して適用する:

$${\color{red}{\int{\frac{f_{1} \tan^{2}{\left(f \right)}}{g} d g}}} = {\color{red}{f_{1} \tan^{2}{\left(f \right)} \int{\frac{1}{g} d g}}}$$

$$$\frac{1}{g}$$$ の不定積分は $$$\int{\frac{1}{g} d g} = \ln{\left(\left|{g}\right| \right)}$$$ です:

$$f_{1} \tan^{2}{\left(f \right)} {\color{red}{\int{\frac{1}{g} d g}}} = f_{1} \tan^{2}{\left(f \right)} {\color{red}{\ln{\left(\left|{g}\right| \right)}}}$$

したがって、

$$\int{\frac{f_{1} \tan^{2}{\left(f \right)}}{g} d g} = f_{1} \ln{\left(\left|{g}\right| \right)} \tan^{2}{\left(f \right)}$$

積分定数を加える:

$$\int{\frac{f_{1} \tan^{2}{\left(f \right)}}{g} d g} = f_{1} \ln{\left(\left|{g}\right| \right)} \tan^{2}{\left(f \right)}+C$$

$$$\int \frac{f_{1} \tan^{2}{\left(f \right)}}{g}\, dg = f_{1} \ln\left(\left|{g}\right|\right) \tan^{2}{\left(f \right)} + C$$$A