Integral of $$$4 - 3 x^{2}$$$

Find $$$\int \left(4 - 3 x^{2}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(4 - 3 x^{2}\right)d x}}} = {\color{red}{\left(\int{4 d x} - \int{3 x^{2} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=4$$$:

$$- \int{3 x^{2} d x} + {\color{red}{\int{4 d x}}} = - \int{3 x^{2} d x} + {\color{red}{\left(4 x\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$4 x - {\color{red}{\int{3 x^{2} d x}}} = 4 x - {\color{red}{\left(3 \int{x^{2} d x}\right)}}$$

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

$$4 x - 3 {\color{red}{\int{x^{2} d x}}}=4 x - 3 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=4 x - 3 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(4 - 3 x^{2}\right)d x} = - x^{3} + 4 x$$

Simplify:

$$\int{\left(4 - 3 x^{2}\right)d x} = x \left(4 - x^{2}\right)$$

Add the constant of integration:

$$\int{\left(4 - 3 x^{2}\right)d x} = x \left(4 - x^{2}\right)+C$$

$$$\int \left(4 - 3 x^{2}\right)\, dx = x \left(4 - x^{2}\right) + C$$$A