Integral dari $$$- \frac{\cos{\left(4 x \right)}}{4}$$$

Temukan $$$\int \left(- \frac{\cos{\left(4 x \right)}}{4}\right)\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=- \frac{1}{4}$$$ dan $$$f{\left(x \right)} = \cos{\left(4 x \right)}$$$:

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

Misalkan $$$u=4 x$$$.

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

Oleh karena itu,

$$- \frac{{\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{4} = - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{4}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ dengan $$$c=\frac{1}{4}$$$ dan $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$- \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{4} = - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{4}$$

Integral dari kosinus adalah $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$- \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{16} = - \frac{{\color{red}{\sin{\left(u \right)}}}}{16}$$

Ingat bahwa $$$u=4 x$$$:

$$- \frac{\sin{\left({\color{red}{u}} \right)}}{16} = - \frac{\sin{\left({\color{red}{\left(4 x\right)}} \right)}}{16}$$

Oleh karena itu,

$$\int{\left(- \frac{\cos{\left(4 x \right)}}{4}\right)d x} = - \frac{\sin{\left(4 x \right)}}{16}$$

Tambahkan konstanta integrasi:

$$\int{\left(- \frac{\cos{\left(4 x \right)}}{4}\right)d x} = - \frac{\sin{\left(4 x \right)}}{16}+C$$

$$$\int \left(- \frac{\cos{\left(4 x \right)}}{4}\right)\, dx = - \frac{\sin{\left(4 x \right)}}{16} + C$$$A