Integral de $$$- \frac{t}{125} + \frac{3 x^{2}}{1000}$$$ em relação a $$$x$$$

Encontre $$$\int \left(- \frac{t}{125} + \frac{3 x^{2}}{1000}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(- \frac{t}{125} + \frac{3 x^{2}}{1000}\right)d x}}} = {\color{red}{\left(- \int{\frac{t}{125} d x} + \int{\frac{3 x^{2}}{1000} d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=\frac{t}{125}$$$:

$$\int{\frac{3 x^{2}}{1000} d x} - {\color{red}{\int{\frac{t}{125} d x}}} = \int{\frac{3 x^{2}}{1000} d x} - {\color{red}{\left(\frac{t x}{125}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{3}{1000}$$$ e $$$f{\left(x \right)} = x^{2}$$$:

$$- \frac{t x}{125} + {\color{red}{\int{\frac{3 x^{2}}{1000} d x}}} = - \frac{t x}{125} + {\color{red}{\left(\frac{3 \int{x^{2} d x}}{1000}\right)}}$$

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

$$- \frac{t x}{125} + \frac{3 {\color{red}{\int{x^{2} d x}}}}{1000}=- \frac{t x}{125} + \frac{3 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{1000}=- \frac{t x}{125} + \frac{3 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{1000}$$

Portanto,

$$\int{\left(- \frac{t}{125} + \frac{3 x^{2}}{1000}\right)d x} = - \frac{t x}{125} + \frac{x^{3}}{1000}$$

Simplifique:

$$\int{\left(- \frac{t}{125} + \frac{3 x^{2}}{1000}\right)d x} = \frac{x \left(- 8 t + x^{2}\right)}{1000}$$

Adicione a constante de integração:

$$\int{\left(- \frac{t}{125} + \frac{3 x^{2}}{1000}\right)d x} = \frac{x \left(- 8 t + x^{2}\right)}{1000}+C$$

$$$\int \left(- \frac{t}{125} + \frac{3 x^{2}}{1000}\right)\, dx = \frac{x \left(- 8 t + x^{2}\right)}{1000} + C$$$A