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

Halla $$$\int \frac{\sin{\left(2 x \right)}}{x}\, dx$$$.

Sea $$$u=2 x$$$.

Entonces $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{2}$$$.

La integral se convierte en

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

Esta integral (Integral seno) no tiene una forma cerrada:

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

Recordemos que $$$u=2 x$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

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