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

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

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

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

Dengan demikian,

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

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

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

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 $$$u=\cos{\left(x \right)}$$$:

$$- \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = - \ln{\left(\left|{{\color{red}{\cos{\left(x \right)}}}}\right| \right)}$$

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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