Integral dari $$$- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}$$$ terhadap $$$x$$$

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

Tulis ulang integran:

$${\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}\right)d x}}} = {\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}}\right)d x}}}$$

Tulis ulang pembilang dan pisahkan pecahan tersebut:

$${\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}}\right)d x}}} = {\color{red}{\int{\left(\frac{\left(- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)}\right) \sin{\left(a \right)}}{\left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right) \left(\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)} + \frac{\cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}\right)d x}}}$$

Integralkan suku demi suku:

$${\color{red}{\int{\left(\frac{\left(- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)}\right) \sin{\left(a \right)}}{\left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right) \left(\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)} + \frac{\cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}\right)d x}}} = {\color{red}{\left(\int{\frac{\cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} d x} + \int{\frac{\left(- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)}\right) \sin{\left(a \right)}}{\left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right) \left(\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)} d x}\right)}}$$

Terapkan aturan konstanta $$$\int c\, dx = c x$$$ dengan $$$c=\frac{\cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}$$$:

$$\int{\frac{\left(- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)}\right) \sin{\left(a \right)}}{\left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right) \left(\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)} d x} + {\color{red}{\int{\frac{\cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} d x}}} = \int{\frac{\left(- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)}\right) \sin{\left(a \right)}}{\left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right) \left(\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)} d x} + {\color{red}{\frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}}}$$

Misalkan $$$u=\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}$$$.

Kemudian $$$du=\left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right)^{\prime }dx = \left(- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)}\right) dx$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$\left(- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)}\right) dx = du$$$.

Integralnya menjadi

$$\frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + {\color{red}{\int{\frac{\left(- \sin{\left(a \right)} \sin{\left(x \right)} - \cos{\left(a \right)} \cos{\left(x \right)}\right) \sin{\left(a \right)}}{\left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right) \left(\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)} d x}}} = \frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + {\color{red}{\int{\frac{\sin{\left(a \right)}}{u \left(\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right)} d u}}}$$

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

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

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

$$\frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + \frac{\sin{\left(a \right)} {\color{red}{\int{\frac{1}{u} d u}}}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} = \frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + \frac{\sin{\left(a \right)} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}$$

Ingat bahwa $$$u=\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}$$$:

$$\frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)} \sin{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} = \frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + \frac{\ln{\left(\left|{{\color{red}{\left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right)}}}\right| \right)} \sin{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}$$

Oleh karena itu,

$$\int{\left(- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}\right)d x} = \frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + \frac{\ln{\left(\left|{\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}}\right| \right)} \sin{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}$$

Sederhanakan:

$$\int{\left(- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}\right)d x} = x \cos{\left(a \right)} + \ln{\left(\left|{\sin{\left(a - x \right)}}\right| \right)} \sin{\left(a \right)}$$

Tambahkan konstanta integrasi:

$$\int{\left(- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}\right)d x} = x \cos{\left(a \right)} + \ln{\left(\left|{\sin{\left(a - x \right)}}\right| \right)} \sin{\left(a \right)}+C$$

$$$\int \left(- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}\right)\, dx = \left(x \cos{\left(a \right)} + \ln\left(\left|{\sin{\left(a - x \right)}}\right|\right) \sin{\left(a \right)}\right) + C$$$A