Integral dari $$$4 \cos{\left(2 x \right)}$$$

Temukan $$$\int 4 \cos{\left(2 x \right)}\, dx$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ dengan $$$c=4$$$ dan $$$f{\left(x \right)} = \cos{\left(2 x \right)}$$$:

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

Misalkan $$$u=2 x$$$.

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

Integral tersebut dapat ditulis ulang sebagai

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

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

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

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

$$2 {\color{red}{\int{\cos{\left(u \right)} d u}}} = 2 {\color{red}{\sin{\left(u \right)}}}$$

Ingat bahwa $$$u=2 x$$$:

$$2 \sin{\left({\color{red}{u}} \right)} = 2 \sin{\left({\color{red}{\left(2 x\right)}} \right)}$$

Oleh karena itu,

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

Tambahkan konstanta integrasi:

$$\int{4 \cos{\left(2 x \right)} d x} = 2 \sin{\left(2 x \right)}+C$$

$$$\int 4 \cos{\left(2 x \right)}\, dx = 2 \sin{\left(2 x \right)} + C$$$A