$$$\frac{\sec{\left(x \right)}}{\sec{\left(2 x \right)}}$$$の積分

$$$\int \frac{\sec{\left(x \right)}}{\sec{\left(2 x \right)}}\, dx$$$ を求めよ。

被積分関数を書き換える:

$${\color{red}{\int{\frac{\sec{\left(x \right)}}{\sec{\left(2 x \right)}} d x}}} = {\color{red}{\int{\frac{2 \cos^{2}{\left(x \right)} - 1}{\cos{\left(x \right)}} d x}}}$$

Expand the expression:

$${\color{red}{\int{\frac{2 \cos^{2}{\left(x \right)} - 1}{\cos{\left(x \right)}} d x}}} = {\color{red}{\int{\left(2 \cos{\left(x \right)} - \frac{1}{\cos{\left(x \right)}}\right)d x}}}$$

項別に積分せよ:

$${\color{red}{\int{\left(2 \cos{\left(x \right)} - \frac{1}{\cos{\left(x \right)}}\right)d x}}} = {\color{red}{\left(- \int{\frac{1}{\cos{\left(x \right)}} d x} + \int{2 \cos{\left(x \right)} d x}\right)}}$$

公式 $$$\cos\left(x\right)=\sin\left(x + \frac{\pi}{2}\right)$$$ を用いて余弦を正弦で表し、次に2倍角の公式 $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$ を用いて正弦を書き換えなさい。:

$$\int{2 \cos{\left(x \right)} d x} - {\color{red}{\int{\frac{1}{\cos{\left(x \right)}} d x}}} = \int{2 \cos{\left(x \right)} d x} - {\color{red}{\int{\frac{1}{2 \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} d x}}}$$

分子と分母に$$$\sec^2\left(\frac{x}{2} + \frac{\pi}{4} \right)$$$を掛ける:

$$\int{2 \cos{\left(x \right)} d x} - {\color{red}{\int{\frac{1}{2 \sin{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} d x}}} = \int{2 \cos{\left(x \right)} d x} - {\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{2 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} d x}}}$$

$$$u=\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$$ とする。

すると $$$du=\left(\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{2} dx$$$（手順は»で確認できます）、$$$\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} dx = 2 du$$$ となります。

この積分は次のように書き換えられる

$$\int{2 \cos{\left(x \right)} d x} - {\color{red}{\int{\frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{2 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} d x}}} = \int{2 \cos{\left(x \right)} d x} - {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$\int{2 \cos{\left(x \right)} d x} - {\color{red}{\int{\frac{1}{u} d u}}} = \int{2 \cos{\left(x \right)} d x} - {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

次のことを思い出してください $$$u=\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$$:

$$- \ln{\left(\left|{{\color{red}{u}}}\right| \right)} + \int{2 \cos{\left(x \right)} d x} = - \ln{\left(\left|{{\color{red}{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}}}\right| \right)} + \int{2 \cos{\left(x \right)} d x}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=2$$$ と $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ に対して適用する:

$$- \ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)} + {\color{red}{\int{2 \cos{\left(x \right)} d x}}} = - \ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)} + {\color{red}{\left(2 \int{\cos{\left(x \right)} d x}\right)}}$$

余弦の積分は$$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$- \ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)} + 2 {\color{red}{\int{\cos{\left(x \right)} d x}}} = - \ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)} + 2 {\color{red}{\sin{\left(x \right)}}}$$

したがって、

$$\int{\frac{\sec{\left(x \right)}}{\sec{\left(2 x \right)}} d x} = - \ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)} + 2 \sin{\left(x \right)}$$

積分定数を加える:

$$\int{\frac{\sec{\left(x \right)}}{\sec{\left(2 x \right)}} d x} = - \ln{\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right| \right)} + 2 \sin{\left(x \right)}+C$$

$$$\int \frac{\sec{\left(x \right)}}{\sec{\left(2 x \right)}}\, dx = \left(- \ln\left(\left|{\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right|\right) + 2 \sin{\left(x \right)}\right) + C$$$A