Integral of $a^{2} t - x^{2}$ with respect to $x$

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

Integrate term by term:

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

Apply the power rule $\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$ $\left(n \neq -1 \right)$ with $n=2$:

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

Apply the constant rule $\int c\, dx = c x$ with $c=a^{2} t$:

$$- \frac{x^{3}}{3} + {\color{red}{\int{a^{2} t d x}}} = - \frac{x^{3}}{3} + {\color{red}{a^{2} t x}}$$

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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