Integral dari $$$\sec{\left(t \right)}$$$

Temukan $$$\int \sec{\left(t \right)}\, dt$$$.

Tulis ulang sekan sebagai $$$\sec\left(t\right)=\frac{1}{\cos\left(t\right)}$$$:

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

Tulis ulang kosinus dalam bentuk sinus menggunakan rumus $$$\cos\left(t\right)=\sin\left(t + \frac{\pi}{2}\right)$$$ dan kemudian tulis ulang sinus menggunakan rumus sudut rangkap $$$\sin\left(t\right)=2\sin\left(\frac{t}{2}\right)\cos\left(\frac{t}{2}\right)$$$:

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

Kalikan pembilang dan penyebut dengan $$$\sec^2\left(\frac{t}{2} + \frac{\pi}{4} \right)$$$:

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

Misalkan $$$u=\tan{\left(\frac{t}{2} + \frac{\pi}{4} \right)}$$$.

Kemudian $$$du=\left(\tan{\left(\frac{t}{2} + \frac{\pi}{4} \right)}\right)^{\prime }dt = \frac{\sec^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{2} dt$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$\sec^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} dt = 2 du$$$.

Jadi,

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

Integral dari $$$\frac{1}{u}$$$ adalah $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Ingat bahwa $$$u=\tan{\left(\frac{t}{2} + \frac{\pi}{4} \right)}$$$:

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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