Integral de $$$t^{2} x - t^{2} y$$$ em relação a $$$x$$$

Encontre $$$\int \left(t^{2} x - t^{2} y\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(t^{2} x - t^{2} y\right)d x}}} = {\color{red}{\left(\int{t^{2} x d x} - \int{t^{2} y d x}\right)}}$$

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

$$- \int{t^{2} y d x} + {\color{red}{\int{t^{2} x d x}}} = - \int{t^{2} y d x} + {\color{red}{t^{2} \int{x d x}}}$$

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

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

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=t^{2} y$$$:

$$\frac{t^{2} x^{2}}{2} - {\color{red}{\int{t^{2} y d x}}} = \frac{t^{2} x^{2}}{2} - {\color{red}{t^{2} x y}}$$

Portanto,

$$\int{\left(t^{2} x - t^{2} y\right)d x} = \frac{t^{2} x^{2}}{2} - t^{2} x y$$

Simplifique:

$$\int{\left(t^{2} x - t^{2} y\right)d x} = \frac{t^{2} x \left(x - 2 y\right)}{2}$$

Adicione a constante de integração:

$$\int{\left(t^{2} x - t^{2} y\right)d x} = \frac{t^{2} x \left(x - 2 y\right)}{2}+C$$

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