Integral de $$$6 e^{- \frac{x}{2}} \sin{\left(2 x \right)}$$$

Encontre $$$\int 6 e^{- \frac{x}{2}} \sin{\left(2 x \right)}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=6$$$ e $$$f{\left(x \right)} = e^{- \frac{x}{2}} \sin{\left(2 x \right)}$$$:

$${\color{red}{\int{6 e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x}}} = {\color{red}{\left(6 \int{e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x}\right)}}$$

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

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

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

Assim,

$$6 {\color{red}{\int{e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x}}}=6 {\color{red}{\left(\sin{\left(2 x \right)} \cdot \left(- 2 e^{- \frac{x}{2}}\right)-\int{\left(- 2 e^{- \frac{x}{2}}\right) \cdot 2 \cos{\left(2 x \right)} d x}\right)}}=6 {\color{red}{\left(- \int{\left(- 4 e^{- \frac{x}{2}} \cos{\left(2 x \right)}\right)d x} - 2 e^{- \frac{x}{2}} \sin{\left(2 x \right)}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=-4$$$ e $$$f{\left(x \right)} = e^{- \frac{x}{2}} \cos{\left(2 x \right)}$$$:

$$- 6 {\color{red}{\int{\left(- 4 e^{- \frac{x}{2}} \cos{\left(2 x \right)}\right)d x}}} - 12 e^{- \frac{x}{2}} \sin{\left(2 x \right)} = - 6 {\color{red}{\left(- 4 \int{e^{- \frac{x}{2}} \cos{\left(2 x \right)} d x}\right)}} - 12 e^{- \frac{x}{2}} \sin{\left(2 x \right)}$$

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

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

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

Logo,

$$24 {\color{red}{\int{e^{- \frac{x}{2}} \cos{\left(2 x \right)} d x}}} - 12 e^{- \frac{x}{2}} \sin{\left(2 x \right)}=24 {\color{red}{\left(\cos{\left(2 x \right)} \cdot \left(- 2 e^{- \frac{x}{2}}\right)-\int{\left(- 2 e^{- \frac{x}{2}}\right) \cdot \left(- 2 \sin{\left(2 x \right)}\right) d x}\right)}} - 12 e^{- \frac{x}{2}} \sin{\left(2 x \right)}=24 {\color{red}{\left(- \int{4 e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x} - 2 e^{- \frac{x}{2}} \cos{\left(2 x \right)}\right)}} - 12 e^{- \frac{x}{2}} \sin{\left(2 x \right)}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=4$$$ e $$$f{\left(x \right)} = e^{- \frac{x}{2}} \sin{\left(2 x \right)}$$$:

$$- 24 {\color{red}{\int{4 e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x}}} - 12 e^{- \frac{x}{2}} \sin{\left(2 x \right)} - 48 e^{- \frac{x}{2}} \cos{\left(2 x \right)} = - 24 {\color{red}{\left(4 \int{e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x}\right)}} - 12 e^{- \frac{x}{2}} \sin{\left(2 x \right)} - 48 e^{- \frac{x}{2}} \cos{\left(2 x \right)}$$

Chegamos a uma integral que já vimos.

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

$$6 \int{e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x} = - 96 \int{e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x} - 12 e^{- \frac{x}{2}} \sin{\left(2 x \right)} - 48 e^{- \frac{x}{2}} \cos{\left(2 x \right)}$$

Resolvendo, obtemos que

$$\int{e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x} = \frac{2 \left(- \sin{\left(2 x \right)} - 4 \cos{\left(2 x \right)}\right) e^{- \frac{x}{2}}}{17}$$

Assim,

$$6 {\color{red}{\int{e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x}}} = 6 {\color{red}{\left(\frac{2 \left(- \sin{\left(2 x \right)} - 4 \cos{\left(2 x \right)}\right) e^{- \frac{x}{2}}}{17}\right)}}$$

Portanto,

$$\int{6 e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x} = \frac{12 \left(- \sin{\left(2 x \right)} - 4 \cos{\left(2 x \right)}\right) e^{- \frac{x}{2}}}{17}$$

Adicione a constante de integração:

$$\int{6 e^{- \frac{x}{2}} \sin{\left(2 x \right)} d x} = \frac{12 \left(- \sin{\left(2 x \right)} - 4 \cos{\left(2 x \right)}\right) e^{- \frac{x}{2}}}{17}+C$$

$$$\int 6 e^{- \frac{x}{2}} \sin{\left(2 x \right)}\, dx = \frac{12 \left(- \sin{\left(2 x \right)} - 4 \cos{\left(2 x \right)}\right) e^{- \frac{x}{2}}}{17} + C$$$A