Integral dari $$$\sin{\left(\frac{3 u}{5} \right)}$$$

Temukan $$$\int \sin{\left(\frac{3 u}{5} \right)}\, du$$$.

Misalkan $$$v=\frac{3 u}{5}$$$.

Kemudian $$$dv=\left(\frac{3 u}{5}\right)^{\prime }du = \frac{3 du}{5}$$$ (langkah-langkah dapat dilihat di »), dan kita memperoleh $$$du = \frac{5 dv}{3}$$$.

Integralnya menjadi

$${\color{red}{\int{\sin{\left(\frac{3 u}{5} \right)} d u}}} = {\color{red}{\int{\frac{5 \sin{\left(v \right)}}{3} d v}}}$$

Terapkan aturan pengali konstanta $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ dengan $$$c=\frac{5}{3}$$$ dan $$$f{\left(v \right)} = \sin{\left(v \right)}$$$:

$${\color{red}{\int{\frac{5 \sin{\left(v \right)}}{3} d v}}} = {\color{red}{\left(\frac{5 \int{\sin{\left(v \right)} d v}}{3}\right)}}$$

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

$$\frac{5 {\color{red}{\int{\sin{\left(v \right)} d v}}}}{3} = \frac{5 {\color{red}{\left(- \cos{\left(v \right)}\right)}}}{3}$$

Ingat bahwa $$$v=\frac{3 u}{5}$$$:

$$- \frac{5 \cos{\left({\color{red}{v}} \right)}}{3} = - \frac{5 \cos{\left({\color{red}{\left(\frac{3 u}{5}\right)}} \right)}}{3}$$

Oleh karena itu,

$$\int{\sin{\left(\frac{3 u}{5} \right)} d u} = - \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3}$$

Tambahkan konstanta integrasi:

$$\int{\sin{\left(\frac{3 u}{5} \right)} d u} = - \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3}+C$$

$$$\int \sin{\left(\frac{3 u}{5} \right)}\, du = - \frac{5 \cos{\left(\frac{3 u}{5} \right)}}{3} + C$$$A