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

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

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,

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

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 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

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

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

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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