$$$\sec{\left(\pi x \right)}$$$ 的積分

求$$$\int \sec{\left(\pi x \right)}\, dx$$$。

令 $$$u=\pi x$$$。

則 $$$du=\left(\pi x\right)^{\prime }dx = \pi dx$$$ (步驟見»)，並可得 $$$dx = \frac{du}{\pi}$$$。

該積分可改寫為

$${\color{red}{\int{\sec{\left(\pi x \right)} d x}}} = {\color{red}{\int{\frac{\sec{\left(u \right)}}{\pi} d u}}}$$

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{1}{\pi}$$$ 與 $$$f{\left(u \right)} = \sec{\left(u \right)}$$$：

$${\color{red}{\int{\frac{\sec{\left(u \right)}}{\pi} d u}}} = {\color{red}{\frac{\int{\sec{\left(u \right)} d u}}{\pi}}}$$

將正割改寫為 $$$\sec\left( u \right)=\frac{1}{\cos\left( u \right)}$$$:

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

使用公式 $$$\cos\left( u \right)=\sin\left( u + \frac{\pi}{2}\right)$$$ 將餘弦用正弦表示，然後使用二倍角公式 $$$\sin\left( u \right)=2\sin\left(\frac{ u }{2}\right)\cos\left(\frac{ u }{2}\right)$$$ 將正弦改寫。:

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

將分子與分母同時乘以 $$$\sec^2\left(\frac{ u }{2} + \frac{\pi}{4} \right)$$$:

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

令 $$$v=\tan{\left(\frac{u}{2} + \frac{\pi}{4} \right)}$$$。

則 $$$dv=\left(\tan{\left(\frac{u}{2} + \frac{\pi}{4} \right)}\right)^{\prime }du = \frac{\sec^{2}{\left(\frac{u}{2} + \frac{\pi}{4} \right)}}{2} du$$$ (步驟見»)，並可得 $$$\sec^{2}{\left(\frac{u}{2} + \frac{\pi}{4} \right)} du = 2 dv$$$。

所以，

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

$$$\frac{1}{v}$$$ 的積分是 $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$：

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

回顧一下 $$$v=\tan{\left(\frac{u}{2} + \frac{\pi}{4} \right)}$$$：

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

回顧一下 $$$u=\pi x$$$：

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

因此，

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

化簡：

$$\int{\sec{\left(\pi x \right)} d x} = \frac{\ln{\left(\left|{\tan{\left(\pi \left(\frac{x}{2} + \frac{1}{4}\right) \right)}}\right| \right)}}{\pi}$$

加上積分常數：

$$\int{\sec{\left(\pi x \right)} d x} = \frac{\ln{\left(\left|{\tan{\left(\pi \left(\frac{x}{2} + \frac{1}{4}\right) \right)}}\right| \right)}}{\pi}+C$$

$$$\int \sec{\left(\pi x \right)}\, dx = \frac{\ln\left(\left|{\tan{\left(\pi \left(\frac{x}{2} + \frac{1}{4}\right) \right)}}\right|\right)}{\pi} + C$$$A