Integral dari $$$2 \cos{\left(\pi t \right)}$$$

Temukan $$$\int 2 \cos{\left(\pi t \right)}\, dt$$$.

Terapkan aturan pengali konstanta $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ dengan $$$c=2$$$ dan $$$f{\left(t \right)} = \cos{\left(\pi t \right)}$$$:

$${\color{red}{\int{2 \cos{\left(\pi t \right)} d t}}} = {\color{red}{\left(2 \int{\cos{\left(\pi t \right)} d t}\right)}}$$

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,

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

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

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

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

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

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

$$\frac{2 \sin{\left({\color{red}{u}} \right)}}{\pi} = \frac{2 \sin{\left({\color{red}{\pi t}} \right)}}{\pi}$$

Oleh karena itu,

$$\int{2 \cos{\left(\pi t \right)} d t} = \frac{2 \sin{\left(\pi t \right)}}{\pi}$$

Tambahkan konstanta integrasi:

$$\int{2 \cos{\left(\pi t \right)} d t} = \frac{2 \sin{\left(\pi t \right)}}{\pi}+C$$

$$$\int 2 \cos{\left(\pi t \right)}\, dt = \frac{2 \sin{\left(\pi t \right)}}{\pi} + C$$$A