Integral of $$$5 - x^{2}$$$

Find $$$\int \left(5 - x^{2}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(5 - x^{2}\right)d x}}} = {\color{red}{\left(\int{5 d x} - \int{x^{2} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=5$$$:

$$- \int{x^{2} d x} + {\color{red}{\int{5 d x}}} = - \int{x^{2} d x} + {\color{red}{\left(5 x\right)}}$$

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

$$5 x - {\color{red}{\int{x^{2} d x}}}=5 x - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=5 x - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(5 - x^{2}\right)d x} = - \frac{x^{3}}{3} + 5 x$$

Simplify:

$$\int{\left(5 - x^{2}\right)d x} = \frac{x \left(15 - x^{2}\right)}{3}$$

Add the constant of integration:

$$\int{\left(5 - x^{2}\right)d x} = \frac{x \left(15 - x^{2}\right)}{3}+C$$

$$$\int \left(5 - x^{2}\right)\, dx = \frac{x \left(15 - x^{2}\right)}{3} + C$$$A