Integral dari $$$\frac{1}{2 \sin{\left(x \right)}}$$$

Temukan $$$\int \frac{1}{2 \sin{\left(x \right)}}\, dx$$$.

Tulis ulang sinus menggunakan rumus sudut ganda $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$:

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

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

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

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

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

Integralnya menjadi

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

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{2}$$$ dan $$$f{\left(u \right)} = \frac{1}{u}$$$:

$${\color{red}{\int{\frac{1}{2 u} d u}}} = {\color{red}{\left(\frac{\int{\frac{1}{u} d u}}{2}\right)}}$$

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

$$\frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2} = \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}$$

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

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

Oleh karena itu,

$$\int{\frac{1}{2 \sin{\left(x \right)}} d x} = \frac{\ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)}}{2}$$

Tambahkan konstanta integrasi:

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

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