Integral de $$$2 x \cos{\left(x^{2} \right)}$$$

Encontre $$$\int 2 x \cos{\left(x^{2} \right)}\, dx$$$.

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

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

Portanto,

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

A integral do cosseno é $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

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

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

$$\sin{\left({\color{red}{u}} \right)} = \sin{\left({\color{red}{x^{2}}} \right)}$$

Portanto,

$$\int{2 x \cos{\left(x^{2} \right)} d x} = \sin{\left(x^{2} \right)}$$

Adicione a constante de integração:

$$\int{2 x \cos{\left(x^{2} \right)} d x} = \sin{\left(x^{2} \right)}+C$$

$$$\int 2 x \cos{\left(x^{2} \right)}\, dx = \sin{\left(x^{2} \right)} + C$$$A