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

Encontre $$$\int \sin{\left(\sqrt{x} \right)}\, dx$$$.

Seja $$$u=\sqrt{x}$$$.

Então $$$du=\left(\sqrt{x}\right)^{\prime }dx = \frac{1}{2 \sqrt{x}} dx$$$ (veja os passos »), e obtemos $$$\frac{dx}{\sqrt{x}} = 2 du$$$.

Portanto,

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

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

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

Para a integral $$$\int{u \sin{\left(u \right)} d u}$$$, use integração por partes $$$\int \operatorname{\kappa} \operatorname{dv} = \operatorname{\kappa}\operatorname{v} - \int \operatorname{v} \operatorname{d\kappa}$$$.

Sejam $$$\operatorname{\kappa}=u$$$ e $$$\operatorname{dv}=\sin{\left(u \right)} du$$$.

Então $$$\operatorname{d\kappa}=\left(u\right)^{\prime }du=1 du$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{\sin{\left(u \right)} d u}=- \cos{\left(u \right)}$$$ (os passos podem ser vistos »).

A integral torna-se

$$2 {\color{red}{\int{u \sin{\left(u \right)} d u}}}=2 {\color{red}{\left(u \cdot \left(- \cos{\left(u \right)}\right)-\int{\left(- \cos{\left(u \right)}\right) \cdot 1 d u}\right)}}=2 {\color{red}{\left(- u \cos{\left(u \right)} - \int{\left(- \cos{\left(u \right)}\right)d u}\right)}}$$

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

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

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

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

Recorde que $$$u=\sqrt{x}$$$:

$$2 \sin{\left({\color{red}{u}} \right)} - 2 {\color{red}{u}} \cos{\left({\color{red}{u}} \right)} = 2 \sin{\left({\color{red}{\sqrt{x}}} \right)} - 2 {\color{red}{\sqrt{x}}} \cos{\left({\color{red}{\sqrt{x}}} \right)}$$

Portanto,

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

Adicione a constante de integração:

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

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