Integraal van $$$t^{2} x - t^{2} y$$$ met betrekking tot $$$x$$$

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

Integreer termgewijs:

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

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=t^{2}$$$ en $$$f{\left(x \right)} = x$$$:

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

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=t^{2} y$$$:

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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