Integral of $$$- 4 x^{2} + \frac{1}{3 \sqrt{x}}$$$

Find $$$\int \left(- 4 x^{2} + \frac{1}{3 \sqrt{x}}\right)\, dx$$$.

Integrate term by term:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = x^{2}$$$:

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

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

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(x \right)} = \frac{1}{\sqrt{x}}$$$:

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

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

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

Therefore,

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

Add the constant of integration:

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

$$$\int \left(- 4 x^{2} + \frac{1}{3 \sqrt{x}}\right)\, dx = \left(\frac{2 \sqrt{x}}{3} - \frac{4 x^{3}}{3}\right) + C$$$A