Integral de $$$\frac{7 d x^{3}}{f} - 13 x^{2} - 6$$$ em relação a $$$x$$$

Encontre $$$\int \left(\frac{7 d x^{3}}{f} - 13 x^{2} - 6\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(\frac{7 d x^{3}}{f} - 13 x^{2} - 6\right)d x}}} = {\color{red}{\left(- \int{6 d x} - \int{13 x^{2} d x} + \int{\frac{7 d x^{3}}{f} d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=6$$$:

$$- \int{13 x^{2} d x} + \int{\frac{7 d x^{3}}{f} d x} - {\color{red}{\int{6 d x}}} = - \int{13 x^{2} d x} + \int{\frac{7 d x^{3}}{f} d x} - {\color{red}{\left(6 x\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=13$$$ e $$$f{\left(x \right)} = x^{2}$$$:

$$- 6 x + \int{\frac{7 d x^{3}}{f} d x} - {\color{red}{\int{13 x^{2} d x}}} = - 6 x + \int{\frac{7 d x^{3}}{f} d x} - {\color{red}{\left(13 \int{x^{2} d x}\right)}}$$

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

$$- 6 x + \int{\frac{7 d x^{3}}{f} d x} - 13 {\color{red}{\int{x^{2} d x}}}=- 6 x + \int{\frac{7 d x^{3}}{f} d x} - 13 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 6 x + \int{\frac{7 d x^{3}}{f} d x} - 13 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

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

$$- \frac{13 x^{3}}{3} - 6 x + {\color{red}{\int{\frac{7 d x^{3}}{f} d x}}} = - \frac{13 x^{3}}{3} - 6 x + {\color{red}{\left(\frac{7 d \int{x^{3} d x}}{f}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=3$$$:

$$\frac{7 d {\color{red}{\int{x^{3} d x}}}}{f} - \frac{13 x^{3}}{3} - 6 x=\frac{7 d {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}}{f} - \frac{13 x^{3}}{3} - 6 x=\frac{7 d {\color{red}{\left(\frac{x^{4}}{4}\right)}}}{f} - \frac{13 x^{3}}{3} - 6 x$$

Portanto,

$$\int{\left(\frac{7 d x^{3}}{f} - 13 x^{2} - 6\right)d x} = \frac{7 d x^{4}}{4 f} - \frac{13 x^{3}}{3} - 6 x$$

Adicione a constante de integração:

$$\int{\left(\frac{7 d x^{3}}{f} - 13 x^{2} - 6\right)d x} = \frac{7 d x^{4}}{4 f} - \frac{13 x^{3}}{3} - 6 x+C$$

$$$\int \left(\frac{7 d x^{3}}{f} - 13 x^{2} - 6\right)\, dx = \left(\frac{7 d x^{4}}{4 f} - \frac{13 x^{3}}{3} - 6 x\right) + C$$$A