$$$\frac{\cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}}{7}$$$ 的积分

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

对 $$$c=\frac{1}{7}$$$ 和 $$$f{\left(x \right)} = \cos{\left(7 x \right)} \sec^{2}{\left(7 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

改写被积函数:

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

使用公式$$$\cos\left(7 x\right)=\sin\left(7 x + \frac{\pi}{2}\right)$$$将余弦用正弦表示，然后使用二倍角公式$$$\sin\left(7 x\right)=2\sin\left(\frac{7 x}{2}\right)\cos\left(\frac{7 x}{2}\right)$$$将正弦改写。:

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

将分子和分母同时乘以 $$$\sec^2\left(\frac{7 x}{2} + \frac{\pi}{4} \right)$$$:

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

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

则$$$du=\left(\tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}\right)^{\prime }dx = \frac{7 \sec^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}}{2} dx$$$ (步骤见»)，并有$$$\sec^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} dx = \frac{2 du}{7}$$$。

积分变为

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

对 $$$c=\frac{1}{7}$$$ 和 $$$f{\left(u \right)} = \frac{1}{u}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

回忆一下 $$$u=\tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}$$$:

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

因此，

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

化简：

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

加上积分常数：

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

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