$$$- \frac{1}{\sin{\left(a - x \right)} \cos{\left(b - x \right)}}$$$ の $$$x$$$ に関する積分

$$$\int \left(- \frac{1}{\sin{\left(a - x \right)} \cos{\left(b - x \right)}}\right)\, dx$$$ を求めよ。

被積分関数を書き換える:

$${\color{red}{\int{\left(- \frac{1}{\sin{\left(a - x \right)} \cos{\left(b - x \right)}}\right)d x}}} = {\color{red}{\int{\left(- \frac{\sin{\left(b - x \right)}}{\cos{\left(a - b \right)} \cos{\left(b - x \right)}} - \frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}}\right)d x}}}$$

項別に積分せよ:

$${\color{red}{\int{\left(- \frac{\sin{\left(b - x \right)}}{\cos{\left(a - b \right)} \cos{\left(b - x \right)}} - \frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}}\right)d x}}} = {\color{red}{\left(- \int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x} - \int{\frac{\sin{\left(b - x \right)}}{\cos{\left(a - b \right)} \cos{\left(b - x \right)}} d x}\right)}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{\cos{\left(a - b \right)}}$$$ と $$$f{\left(x \right)} = \frac{\sin{\left(b - x \right)}}{\cos{\left(b - x \right)}}$$$ に対して適用する:

$$- \int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x} - {\color{red}{\int{\frac{\sin{\left(b - x \right)}}{\cos{\left(a - b \right)} \cos{\left(b - x \right)}} d x}}} = - \int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x} - {\color{red}{\frac{\int{\frac{\sin{\left(b - x \right)}}{\cos{\left(b - x \right)}} d x}}{\cos{\left(a - b \right)}}}}$$

$$$u=\cos{\left(b - x \right)}$$$ とする。

すると $$$du=\left(\cos{\left(b - x \right)}\right)^{\prime }dx = \sin{\left(b - x \right)} dx$$$（手順は»で確認できます）、$$$\sin{\left(b - x \right)} dx = du$$$ となります。

したがって、

$$- \int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x} - \frac{{\color{red}{\int{\frac{\sin{\left(b - x \right)}}{\cos{\left(b - x \right)}} d x}}}}{\cos{\left(a - b \right)}} = - \int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x} - \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{\cos{\left(a - b \right)}}$$

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

$$- \int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x} - \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{\cos{\left(a - b \right)}} = - \int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x} - \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{\cos{\left(a - b \right)}}$$

次のことを思い出してください $$$u=\cos{\left(b - x \right)}$$$:

$$- \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{\cos{\left(a - b \right)}} - \int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x} = - \frac{\ln{\left(\left|{{\color{red}{\cos{\left(b - x \right)}}}}\right| \right)}}{\cos{\left(a - b \right)}} - \int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{\cos{\left(a - b \right)}}$$$ と $$$f{\left(x \right)} = \frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)}}$$$ に対して適用する:

$$- \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} - {\color{red}{\int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)} \cos{\left(a - b \right)}} d x}}} = - \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} - {\color{red}{\frac{\int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)}} d x}}{\cos{\left(a - b \right)}}}}$$

$$$u=\sin{\left(a - x \right)}$$$ とする。

すると $$$du=\left(\sin{\left(a - x \right)}\right)^{\prime }dx = - \cos{\left(a - x \right)} dx$$$（手順は»で確認できます）、$$$\cos{\left(a - x \right)} dx = - du$$$ となります。

この積分は次のように書き換えられる

$$- \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} - \frac{{\color{red}{\int{\frac{\cos{\left(a - x \right)}}{\sin{\left(a - x \right)}} d x}}}}{\cos{\left(a - b \right)}} = - \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} - \frac{{\color{red}{\int{\left(- \frac{1}{u}\right)d u}}}}{\cos{\left(a - b \right)}}$$

定数倍の法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ を、$$$c=-1$$$ と $$$f{\left(u \right)} = \frac{1}{u}$$$ に対して適用する:

$$- \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} - \frac{{\color{red}{\int{\left(- \frac{1}{u}\right)d u}}}}{\cos{\left(a - b \right)}} = - \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} - \frac{{\color{red}{\left(- \int{\frac{1}{u} d u}\right)}}}{\cos{\left(a - b \right)}}$$

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

$$- \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{\cos{\left(a - b \right)}} = - \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} + \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{\cos{\left(a - b \right)}}$$

次のことを思い出してください $$$u=\sin{\left(a - x \right)}$$$:

$$- \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} + \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{\cos{\left(a - b \right)}} = - \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} + \frac{\ln{\left(\left|{{\color{red}{\sin{\left(a - x \right)}}}}\right| \right)}}{\cos{\left(a - b \right)}}$$

したがって、

$$\int{\left(- \frac{1}{\sin{\left(a - x \right)} \cos{\left(b - x \right)}}\right)d x} = \frac{\ln{\left(\left|{\sin{\left(a - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}} - \frac{\ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}}$$

簡単化せよ:

$$\int{\left(- \frac{1}{\sin{\left(a - x \right)} \cos{\left(b - x \right)}}\right)d x} = \frac{\ln{\left(\left|{\sin{\left(a - x \right)}}\right| \right)} - \ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}}$$

積分定数を加える:

$$\int{\left(- \frac{1}{\sin{\left(a - x \right)} \cos{\left(b - x \right)}}\right)d x} = \frac{\ln{\left(\left|{\sin{\left(a - x \right)}}\right| \right)} - \ln{\left(\left|{\cos{\left(b - x \right)}}\right| \right)}}{\cos{\left(a - b \right)}}+C$$

$$$\int \left(- \frac{1}{\sin{\left(a - x \right)} \cos{\left(b - x \right)}}\right)\, dx = \frac{\ln\left(\left|{\sin{\left(a - x \right)}}\right|\right) - \ln\left(\left|{\cos{\left(b - x \right)}}\right|\right)}{\cos{\left(a - b \right)}} + C$$$A