Integral de $$$28 x \sin{\left(3 \right)} \cos{\left(7 x \right)}$$$

Encontre $$$\int 28 x \sin{\left(3 \right)} \cos{\left(7 x \right)}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=28 \sin{\left(3 \right)}$$$ e $$$f{\left(x \right)} = x \cos{\left(7 x \right)}$$$:

$${\color{red}{\int{28 x \sin{\left(3 \right)} \cos{\left(7 x \right)} d x}}} = {\color{red}{\left(28 \sin{\left(3 \right)} \int{x \cos{\left(7 x \right)} d x}\right)}}$$

Para a integral $$$\int{x \cos{\left(7 x \right)} d x}$$$, use integração por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sejam $$$\operatorname{u}=x$$$ e $$$\operatorname{dv}=\cos{\left(7 x \right)} dx$$$.

Então $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{\cos{\left(7 x \right)} d x}=\frac{\sin{\left(7 x \right)}}{7}$$$ (os passos podem ser vistos »).

A integral torna-se

$$28 \sin{\left(3 \right)} {\color{red}{\int{x \cos{\left(7 x \right)} d x}}}=28 \sin{\left(3 \right)} {\color{red}{\left(x \cdot \frac{\sin{\left(7 x \right)}}{7}-\int{\frac{\sin{\left(7 x \right)}}{7} \cdot 1 d x}\right)}}=28 \sin{\left(3 \right)} {\color{red}{\left(\frac{x \sin{\left(7 x \right)}}{7} - \int{\frac{\sin{\left(7 x \right)}}{7} d x}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{7}$$$ e $$$f{\left(x \right)} = \sin{\left(7 x \right)}$$$:

$$28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} - {\color{red}{\int{\frac{\sin{\left(7 x \right)}}{7} d x}}}\right) = 28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} - {\color{red}{\left(\frac{\int{\sin{\left(7 x \right)} d x}}{7}\right)}}\right)$$

Seja $$$u=7 x$$$.

Então $$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{7}$$$.

Portanto,

$$28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} - \frac{{\color{red}{\int{\sin{\left(7 x \right)} d x}}}}{7}\right) = 28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{7} d u}}}}{7}\right)$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{7}$$$ e $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{7} d u}}}}{7}\right) = 28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} - \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{7}\right)}}}{7}\right)$$

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

$$28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{49}\right) = 28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{49}\right)$$

Recorde que $$$u=7 x$$$:

$$28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} + \frac{\cos{\left({\color{red}{u}} \right)}}{49}\right) = 28 \sin{\left(3 \right)} \left(\frac{x \sin{\left(7 x \right)}}{7} + \frac{\cos{\left({\color{red}{\left(7 x\right)}} \right)}}{49}\right)$$

Portanto,

$$\int{28 x \sin{\left(3 \right)} \cos{\left(7 x \right)} d x} = 28 \left(\frac{x \sin{\left(7 x \right)}}{7} + \frac{\cos{\left(7 x \right)}}{49}\right) \sin{\left(3 \right)}$$

Simplifique:

$$\int{28 x \sin{\left(3 \right)} \cos{\left(7 x \right)} d x} = \frac{4 \left(7 x \sin{\left(7 x \right)} + \cos{\left(7 x \right)}\right) \sin{\left(3 \right)}}{7}$$

Adicione a constante de integração:

$$\int{28 x \sin{\left(3 \right)} \cos{\left(7 x \right)} d x} = \frac{4 \left(7 x \sin{\left(7 x \right)} + \cos{\left(7 x \right)}\right) \sin{\left(3 \right)}}{7}+C$$

$$$\int 28 x \sin{\left(3 \right)} \cos{\left(7 x \right)}\, dx = \frac{4 \left(7 x \sin{\left(7 x \right)} + \cos{\left(7 x \right)}\right) \sin{\left(3 \right)}}{7} + C$$$A