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

Encontre $$$\int \frac{\sin{\left(t^{2} \right)}}{t}\, dt$$$.

Seja $$$u=t^{2}$$$.

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

A integral pode ser reescrita como

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(u \right)} = \frac{\sin{\left(u \right)}}{u}$$$:

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

Esta integral (Integral seno) não possui forma fechada:

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

Recorde que $$$u=t^{2}$$$:

$$\frac{\operatorname{Si}{\left({\color{red}{u}} \right)}}{2} = \frac{\operatorname{Si}{\left({\color{red}{t^{2}}} \right)}}{2}$$

Portanto,

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

Adicione a constante de integração:

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

$$$\int \frac{\sin{\left(t^{2} \right)}}{t}\, dt = \frac{\operatorname{Si}{\left(t^{2} \right)}}{2} + C$$$A