Integral of $$$1 - u^{2}$$$

Find $$$\int \left(1 - u^{2}\right)\, du$$$.

Integrate term by term:

$${\color{red}{\int{\left(1 - u^{2}\right)d u}}} = {\color{red}{\left(\int{1 d u} - \int{u^{2} d u}\right)}}$$

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=1$$$:

$$- \int{u^{2} d u} + {\color{red}{\int{1 d u}}} = - \int{u^{2} d u} + {\color{red}{u}}$$

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

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

Therefore,

$$\int{\left(1 - u^{2}\right)d u} = - \frac{u^{3}}{3} + u$$

Add the constant of integration:

$$\int{\left(1 - u^{2}\right)d u} = - \frac{u^{3}}{3} + u+C$$

$$$\int \left(1 - u^{2}\right)\, du = \left(- \frac{u^{3}}{3} + u\right) + C$$$A