Integral de $$$- \frac{x}{2} - \sin{\left(e x \right)} + \cos{\left(x_{1} \right)}$$$ em relação a $$$x$$$

Encontre $$$\int \left(- \frac{x}{2} - \sin{\left(e x \right)} + \cos{\left(x_{1} \right)}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(- \frac{x}{2} - \sin{\left(e x \right)} + \cos{\left(x_{1} \right)}\right)d x}}} = {\color{red}{\left(- \int{\frac{x}{2} d x} - \int{\sin{\left(e x \right)} d x} + \int{\cos{\left(x_{1} \right)} d x}\right)}}$$

Seja $$$u=e x$$$.

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

Portanto,

$$- \int{\frac{x}{2} d x} + \int{\cos{\left(x_{1} \right)} d x} - {\color{red}{\int{\sin{\left(e x \right)} d x}}} = - \int{\frac{x}{2} d x} + \int{\cos{\left(x_{1} \right)} d x} - {\color{red}{\int{\frac{\sin{\left(u \right)}}{e} d u}}}$$

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

$$- \int{\frac{x}{2} d x} + \int{\cos{\left(x_{1} \right)} d x} - {\color{red}{\int{\frac{\sin{\left(u \right)}}{e} d u}}} = - \int{\frac{x}{2} d x} + \int{\cos{\left(x_{1} \right)} d x} - {\color{red}{\frac{\int{\sin{\left(u \right)} d u}}{e}}}$$

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

$$- \int{\frac{x}{2} d x} + \int{\cos{\left(x_{1} \right)} d x} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{e} = - \int{\frac{x}{2} d x} + \int{\cos{\left(x_{1} \right)} d x} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{e}$$

Recorde que $$$u=e x$$$:

$$- \int{\frac{x}{2} d x} + \int{\cos{\left(x_{1} \right)} d x} + \frac{\cos{\left({\color{red}{u}} \right)}}{e} = - \int{\frac{x}{2} d x} + \int{\cos{\left(x_{1} \right)} d x} + \frac{\cos{\left({\color{red}{e x}} \right)}}{e}$$

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 $$$f{\left(x \right)} = x$$$:

$$\frac{\cos{\left(e x \right)}}{e} + \int{\cos{\left(x_{1} \right)} d x} - {\color{red}{\int{\frac{x}{2} d x}}} = \frac{\cos{\left(e x \right)}}{e} + \int{\cos{\left(x_{1} \right)} d x} - {\color{red}{\left(\frac{\int{x d x}}{2}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=1$$$:

$$\frac{\cos{\left(e x \right)}}{e} + \int{\cos{\left(x_{1} \right)} d x} - \frac{{\color{red}{\int{x d x}}}}{2}=\frac{\cos{\left(e x \right)}}{e} + \int{\cos{\left(x_{1} \right)} d x} - \frac{{\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{2}=\frac{\cos{\left(e x \right)}}{e} + \int{\cos{\left(x_{1} \right)} d x} - \frac{{\color{red}{\left(\frac{x^{2}}{2}\right)}}}{2}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=\cos{\left(x_{1} \right)}$$$:

$$- \frac{x^{2}}{4} + \frac{\cos{\left(e x \right)}}{e} + {\color{red}{\int{\cos{\left(x_{1} \right)} d x}}} = - \frac{x^{2}}{4} + \frac{\cos{\left(e x \right)}}{e} + {\color{red}{x \cos{\left(x_{1} \right)}}}$$

Portanto,

$$\int{\left(- \frac{x}{2} - \sin{\left(e x \right)} + \cos{\left(x_{1} \right)}\right)d x} = - \frac{x^{2}}{4} + x \cos{\left(x_{1} \right)} + \frac{\cos{\left(e x \right)}}{e}$$

Adicione a constante de integração:

$$\int{\left(- \frac{x}{2} - \sin{\left(e x \right)} + \cos{\left(x_{1} \right)}\right)d x} = - \frac{x^{2}}{4} + x \cos{\left(x_{1} \right)} + \frac{\cos{\left(e x \right)}}{e}+C$$

$$$\int \left(- \frac{x}{2} - \sin{\left(e x \right)} + \cos{\left(x_{1} \right)}\right)\, dx = \left(- \frac{x^{2}}{4} + x \cos{\left(x_{1} \right)} + \frac{\cos{\left(e x \right)}}{e}\right) + C$$$A