Integral de $$$\frac{\sin{\left(2 z \right)}}{z}$$$

Encontre $$$\int \frac{\sin{\left(2 z \right)}}{z}\, dz$$$.

Seja $$$u=2 z$$$.

Então $$$du=\left(2 z\right)^{\prime }dz = 2 dz$$$ (veja os passos »), e obtemos $$$dz = \frac{du}{2}$$$.

Portanto,

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

Esta integral (Integral seno) não possui forma fechada:

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

Recorde que $$$u=2 z$$$:

$$\operatorname{Si}{\left({\color{red}{u}} \right)} = \operatorname{Si}{\left({\color{red}{\left(2 z\right)}} \right)}$$

Portanto,

$$\int{\frac{\sin{\left(2 z \right)}}{z} d z} = \operatorname{Si}{\left(2 z \right)}$$

Adicione a constante de integração:

$$\int{\frac{\sin{\left(2 z \right)}}{z} d z} = \operatorname{Si}{\left(2 z \right)}+C$$

$$$\int \frac{\sin{\left(2 z \right)}}{z}\, dz = \operatorname{Si}{\left(2 z \right)} + C$$$A