Integral dari $$$\cos{\left(\frac{t}{a} \right)}$$$ terhadap $$$t$$$

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

Misalkan $$$u=\frac{t}{a}$$$.

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

Integral tersebut dapat ditulis ulang sebagai

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

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

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

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

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

Ingat bahwa $$$u=\frac{t}{a}$$$:

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

Oleh karena itu,

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

Tambahkan konstanta integrasi:

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

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