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

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

Para a integral $$$\int{e^{x} \sin{\left(\frac{x}{2} \right)} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=\sin{\left(\frac{x}{2} \right)}$$$ e $$$\operatorname{dv}=e^{x} dx$$$.

Então $$$\operatorname{du}=\left(\sin{\left(\frac{x}{2} \right)}\right)^{\prime }dx=\frac{\cos{\left(\frac{x}{2} \right)}}{2} dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{e^{x} d x}=e^{x}$$$ (os passos podem ser vistos »).

A integral pode ser reescrita como

$${\color{red}{\int{e^{x} \sin{\left(\frac{x}{2} \right)} d x}}}={\color{red}{\left(\sin{\left(\frac{x}{2} \right)} \cdot e^{x}-\int{e^{x} \cdot \frac{\cos{\left(\frac{x}{2} \right)}}{2} d x}\right)}}={\color{red}{\left(e^{x} \sin{\left(\frac{x}{2} \right)} - \int{\frac{e^{x} \cos{\left(\frac{x}{2} \right)}}{2} d x}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(x \right)} = e^{x} \cos{\left(\frac{x}{2} \right)}$$$:

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

Para a integral $$$\int{e^{x} \cos{\left(\frac{x}{2} \right)} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=\cos{\left(\frac{x}{2} \right)}$$$ e $$$\operatorname{dv}=e^{x} dx$$$.

Então $$$\operatorname{du}=\left(\cos{\left(\frac{x}{2} \right)}\right)^{\prime }dx=- \frac{\sin{\left(\frac{x}{2} \right)}}{2} dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{e^{x} d x}=e^{x}$$$ (os passos podem ser vistos »).

Assim,

$$e^{x} \sin{\left(\frac{x}{2} \right)} - \frac{{\color{red}{\int{e^{x} \cos{\left(\frac{x}{2} \right)} d x}}}}{2}=e^{x} \sin{\left(\frac{x}{2} \right)} - \frac{{\color{red}{\left(\cos{\left(\frac{x}{2} \right)} \cdot e^{x}-\int{e^{x} \cdot \left(- \frac{\sin{\left(\frac{x}{2} \right)}}{2}\right) d x}\right)}}}{2}=e^{x} \sin{\left(\frac{x}{2} \right)} - \frac{{\color{red}{\left(e^{x} \cos{\left(\frac{x}{2} \right)} - \int{\left(- \frac{e^{x} \sin{\left(\frac{x}{2} \right)}}{2}\right)d x}\right)}}}{2}$$

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

$$e^{x} \sin{\left(\frac{x}{2} \right)} - \frac{e^{x} \cos{\left(\frac{x}{2} \right)}}{2} + \frac{{\color{red}{\int{\left(- \frac{e^{x} \sin{\left(\frac{x}{2} \right)}}{2}\right)d x}}}}{2} = e^{x} \sin{\left(\frac{x}{2} \right)} - \frac{e^{x} \cos{\left(\frac{x}{2} \right)}}{2} + \frac{{\color{red}{\left(- \frac{\int{e^{x} \sin{\left(\frac{x}{2} \right)} d x}}{2}\right)}}}{2}$$

Chegamos a uma integral que já vimos.

Assim, obtivemos a seguinte equação simples em relação à integral:

$$\int{e^{x} \sin{\left(\frac{x}{2} \right)} d x} = e^{x} \sin{\left(\frac{x}{2} \right)} - \frac{e^{x} \cos{\left(\frac{x}{2} \right)}}{2} - \frac{\int{e^{x} \sin{\left(\frac{x}{2} \right)} d x}}{4}$$

Resolvendo, obtemos que

$$\int{e^{x} \sin{\left(\frac{x}{2} \right)} d x} = \frac{2 \left(2 \sin{\left(\frac{x}{2} \right)} - \cos{\left(\frac{x}{2} \right)}\right) e^{x}}{5}$$

Portanto,

$$\int{e^{x} \sin{\left(\frac{x}{2} \right)} d x} = \frac{2 \left(2 \sin{\left(\frac{x}{2} \right)} - \cos{\left(\frac{x}{2} \right)}\right) e^{x}}{5}$$

Adicione a constante de integração:

$$\int{e^{x} \sin{\left(\frac{x}{2} \right)} d x} = \frac{2 \left(2 \sin{\left(\frac{x}{2} \right)} - \cos{\left(\frac{x}{2} \right)}\right) e^{x}}{5}+C$$

$$$\int e^{x} \sin{\left(\frac{x}{2} \right)}\, dx = \frac{2 \left(2 \sin{\left(\frac{x}{2} \right)} - \cos{\left(\frac{x}{2} \right)}\right) e^{x}}{5} + C$$$A