$$$\frac{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}}{7}$$$の積分

$$$\int \frac{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}}{7}\, dx$$$ を求めよ。

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{7}$$$ と $$$f{\left(x \right)} = \cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}$$$ に対して適用する:

$${\color{red}{\int{\frac{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}}{7} d x}}} = {\color{red}{\left(\frac{\int{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)} d x}}{7}\right)}}$$

被積分関数を書き換える:

$$\frac{{\color{red}{\int{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)} d x}}}}{7} = \frac{{\color{red}{\int{\frac{1}{\cos{\left(7 x \right)}} d x}}}}{7}$$

公式 $$$\cos\left(7 x\right)=\sin\left(7 x + \frac{\pi}{2}\right)$$$ を用いて余弦を正弦で表し、次に2倍角の公式 $$$\sin\left(7 x\right)=2\sin\left(\frac{7 x}{2}\right)\cos\left(\frac{7 x}{2}\right)$$$ を用いて正弦を書き換えなさい。:

$$\frac{{\color{red}{\int{\frac{1}{\cos{\left(7 x \right)}} d x}}}}{7} = \frac{{\color{red}{\int{\frac{1}{2 \sin{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}} d x}}}}{7}$$

分子と分母に$$$\sec^2\left(\frac{7 x}{2} + \frac{\pi}{4} \right)$$$を掛ける:

$$\frac{{\color{red}{\int{\frac{1}{2 \sin{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} \cos{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}} d x}}}}{7} = \frac{{\color{red}{\int{\frac{\sec^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}}{2 \tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}} d x}}}}{7}$$

$$$u=\tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}$$$ とする。

すると $$$du=\left(\tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}\right)^{\prime }dx = \frac{7 \sec^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}}{2} dx$$$（手順は»で確認できます）、$$$\sec^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} dx = \frac{2 du}{7}$$$ となります。

したがって、

$$\frac{{\color{red}{\int{\frac{\sec^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}}{2 \tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}} d x}}}}{7} = \frac{{\color{red}{\int{\frac{1}{7 u} d u}}}}{7}$$

定数倍の法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ を、$$$c=\frac{1}{7}$$$ と $$$f{\left(u \right)} = \frac{1}{u}$$$ に対して適用する:

$$\frac{{\color{red}{\int{\frac{1}{7 u} d u}}}}{7} = \frac{{\color{red}{\left(\frac{\int{\frac{1}{u} d u}}{7}\right)}}}{7}$$

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

$$\frac{{\color{red}{\int{\frac{1}{u} d u}}}}{49} = \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{49}$$

次のことを思い出してください $$$u=\tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}$$$:

$$\frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{49} = \frac{\ln{\left(\left|{{\color{red}{\tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}}}}\right| \right)}}{49}$$

したがって、

$$\int{\frac{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}}{7} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}}\right| \right)}}{49}$$

簡単化せよ:

$$\int{\frac{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}}{7} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{14 x + \pi}{4} \right)}}\right| \right)}}{49}$$

積分定数を加える:

$$\int{\frac{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}}{7} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{14 x + \pi}{4} \right)}}\right| \right)}}{49}+C$$

$$$\int \frac{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}}{7}\, dx = \frac{\ln\left(\left|{\tan{\left(\frac{14 x + \pi}{4} \right)}}\right|\right)}{49} + C$$$A