Integral dari $$$\cos{\left(\pi n x \right)}$$$ terhadap $$$x$$$

Temukan $$$\int \cos{\left(\pi n x \right)}\, dx$$$.

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

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

Jadi,

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

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

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

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

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

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

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

Oleh karena itu,

$$\int{\cos{\left(\pi n x \right)} d x} = \frac{\sin{\left(\pi n x \right)}}{\pi n}$$

Tambahkan konstanta integrasi:

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

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