Integral de $$$a l t \left(x - \pi\right) \cos{\left(x \right)}$$$ em relação a $$$x$$$

Encontre $$$\int a l t \left(x - \pi\right) \cos{\left(x \right)}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=a l t$$$ e $$$f{\left(x \right)} = \left(x - \pi\right) \cos{\left(x \right)}$$$:

$${\color{red}{\int{a l t \left(x - \pi\right) \cos{\left(x \right)} d x}}} = {\color{red}{a l t \int{\left(x - \pi\right) \cos{\left(x \right)} d x}}}$$

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

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

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

Assim,

$$a l t {\color{red}{\int{\left(x - \pi\right) \cos{\left(x \right)} d x}}}=a l t {\color{red}{\left(\left(x - \pi\right) \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 1 d x}\right)}}=a l t {\color{red}{\left(\left(x - \pi\right) \sin{\left(x \right)} - \int{\sin{\left(x \right)} d x}\right)}}$$

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

$$a l t \left(\left(x - \pi\right) \sin{\left(x \right)} - {\color{red}{\int{\sin{\left(x \right)} d x}}}\right) = a l t \left(\left(x - \pi\right) \sin{\left(x \right)} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}\right)$$

Portanto,

$$\int{a l t \left(x - \pi\right) \cos{\left(x \right)} d x} = a l t \left(\left(x - \pi\right) \sin{\left(x \right)} + \cos{\left(x \right)}\right)$$

Simplifique:

$$\int{a l t \left(x - \pi\right) \cos{\left(x \right)} d x} = - a l t \left(\left(\pi - x\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right)$$

Adicione a constante de integração:

$$\int{a l t \left(x - \pi\right) \cos{\left(x \right)} d x} = - a l t \left(\left(\pi - x\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right)+C$$

$$$\int a l t \left(x - \pi\right) \cos{\left(x \right)}\, dx = - a l t \left(\left(\pi - x\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right) + C$$$A