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

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

Integrate term by term:

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

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

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

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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