Integral dari $$$\cos{\left(\pi t \right)}$$$

Temukan $$$\int \cos{\left(\pi t \right)}\, dt$$$.

Misalkan $$$u=\pi t$$$.

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

Jadi,

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

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

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

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

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

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

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

$$$\int \cos{\left(\pi t \right)}\, dt = \frac{\sin{\left(\pi t \right)}}{\pi} + C$$$A