Integral dari $$$4 \pi \sin{\left(\pi x \right)}$$$

Temukan $$$\int 4 \pi \sin{\left(\pi x \right)}\, dx$$$.

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

$${\color{red}{\int{4 \pi \sin{\left(\pi x \right)} d x}}} = {\color{red}{\left(4 \pi \int{\sin{\left(\pi x \right)} d x}\right)}}$$

Misalkan $$$u=\pi x$$$.

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

Jadi,

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

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

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

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

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

Ingat bahwa $$$u=\pi x$$$:

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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