Integral de $$$2 x - 2 - \frac{7 \sqrt{3}}{3 \sqrt{x}}$$$

Encontre $$$\int \left(2 x - 2 - \frac{7 \sqrt{3}}{3 \sqrt{x}}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(2 x - 2 - \frac{7 \sqrt{3}}{3 \sqrt{x}}\right)d x}}} = {\color{red}{\left(- \int{2 d x} + \int{2 x d x} - \int{\frac{7 \sqrt{3}}{3 \sqrt{x}} d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=2$$$:

$$\int{2 x d x} - \int{\frac{7 \sqrt{3}}{3 \sqrt{x}} d x} - {\color{red}{\int{2 d x}}} = \int{2 x d x} - \int{\frac{7 \sqrt{3}}{3 \sqrt{x}} d x} - {\color{red}{\left(2 x\right)}}$$

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

$$- 2 x - \int{\frac{7 \sqrt{3}}{3 \sqrt{x}} d x} + {\color{red}{\int{2 x d x}}} = - 2 x - \int{\frac{7 \sqrt{3}}{3 \sqrt{x}} d x} + {\color{red}{\left(2 \int{x d x}\right)}}$$

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

$$- 2 x - \int{\frac{7 \sqrt{3}}{3 \sqrt{x}} d x} + 2 {\color{red}{\int{x d x}}}=- 2 x - \int{\frac{7 \sqrt{3}}{3 \sqrt{x}} d x} + 2 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 2 x - \int{\frac{7 \sqrt{3}}{3 \sqrt{x}} d x} + 2 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{7 \sqrt{3}}{3}$$$ e $$$f{\left(x \right)} = \frac{1}{\sqrt{x}}$$$:

$$x^{2} - 2 x - {\color{red}{\int{\frac{7 \sqrt{3}}{3 \sqrt{x}} d x}}} = x^{2} - 2 x - {\color{red}{\left(\frac{7 \sqrt{3} \int{\frac{1}{\sqrt{x}} d x}}{3}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=- \frac{1}{2}$$$:

$$x^{2} - 2 x - \frac{7 \sqrt{3} {\color{red}{\int{\frac{1}{\sqrt{x}} d x}}}}{3}=x^{2} - 2 x - \frac{7 \sqrt{3} {\color{red}{\int{x^{- \frac{1}{2}} d x}}}}{3}=x^{2} - 2 x - \frac{7 \sqrt{3} {\color{red}{\frac{x^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{3}=x^{2} - 2 x - \frac{7 \sqrt{3} {\color{red}{\left(2 x^{\frac{1}{2}}\right)}}}{3}=x^{2} - 2 x - \frac{7 \sqrt{3} {\color{red}{\left(2 \sqrt{x}\right)}}}{3}$$

Portanto,

$$\int{\left(2 x - 2 - \frac{7 \sqrt{3}}{3 \sqrt{x}}\right)d x} = - \frac{14 \sqrt{3} \sqrt{x}}{3} + x^{2} - 2 x$$

Adicione a constante de integração:

$$\int{\left(2 x - 2 - \frac{7 \sqrt{3}}{3 \sqrt{x}}\right)d x} = - \frac{14 \sqrt{3} \sqrt{x}}{3} + x^{2} - 2 x+C$$

$$$\int \left(2 x - 2 - \frac{7 \sqrt{3}}{3 \sqrt{x}}\right)\, dx = \left(- \frac{14 \sqrt{3} \sqrt{x}}{3} + x^{2} - 2 x\right) + C$$$A