Integral dari $$$\frac{y}{\cos{\left(x \right)}}$$$ terhadap $$$x$$$

Temukan $$$\int \frac{y}{\cos{\left(x \right)}}\, dx$$$.

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

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

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

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

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

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

Jadi,

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

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=y$$$ dan $$$f{\left(u \right)} = \frac{1}{u}$$$:

$${\color{red}{\int{\frac{y}{u} d u}}} = {\color{red}{y \int{\frac{1}{u} d u}}}$$

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

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

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

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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