Integral of $$$- 4 x^{4} + 5 x^{2} - 5 x - 2$$$

Find $$$\int \left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)d x}}} = {\color{red}{\left(- \int{2 d x} - \int{5 x d x} + \int{5 x^{2} d x} - \int{4 x^{4} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=2$$$:

$$- \int{5 x d x} + \int{5 x^{2} d x} - \int{4 x^{4} d x} - {\color{red}{\int{2 d x}}} = - \int{5 x d x} + \int{5 x^{2} d x} - \int{4 x^{4} d x} - {\color{red}{\left(2 x\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=5$$$ and $$$f{\left(x \right)} = x$$$:

$$- 2 x + \int{5 x^{2} d x} - \int{4 x^{4} d x} - {\color{red}{\int{5 x d x}}} = - 2 x + \int{5 x^{2} d x} - \int{4 x^{4} d x} - {\color{red}{\left(5 \int{x d x}\right)}}$$

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

$$- 2 x + \int{5 x^{2} d x} - \int{4 x^{4} d x} - 5 {\color{red}{\int{x d x}}}=- 2 x + \int{5 x^{2} d x} - \int{4 x^{4} d x} - 5 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 2 x + \int{5 x^{2} d x} - \int{4 x^{4} d x} - 5 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

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

$$- \frac{5 x^{2}}{2} - 2 x + \int{5 x^{2} d x} - {\color{red}{\int{4 x^{4} d x}}} = - \frac{5 x^{2}}{2} - 2 x + \int{5 x^{2} d x} - {\color{red}{\left(4 \int{x^{4} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$- \frac{5 x^{2}}{2} - 2 x + \int{5 x^{2} d x} - 4 {\color{red}{\int{x^{4} d x}}}=- \frac{5 x^{2}}{2} - 2 x + \int{5 x^{2} d x} - 4 {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=- \frac{5 x^{2}}{2} - 2 x + \int{5 x^{2} d x} - 4 {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=5$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$- \frac{4 x^{5}}{5} - \frac{5 x^{2}}{2} - 2 x + {\color{red}{\int{5 x^{2} d x}}} = - \frac{4 x^{5}}{5} - \frac{5 x^{2}}{2} - 2 x + {\color{red}{\left(5 \int{x^{2} d x}\right)}}$$

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

$$- \frac{4 x^{5}}{5} - \frac{5 x^{2}}{2} - 2 x + 5 {\color{red}{\int{x^{2} d x}}}=- \frac{4 x^{5}}{5} - \frac{5 x^{2}}{2} - 2 x + 5 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \frac{4 x^{5}}{5} - \frac{5 x^{2}}{2} - 2 x + 5 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)d x} = - \frac{4 x^{5}}{5} + \frac{5 x^{3}}{3} - \frac{5 x^{2}}{2} - 2 x$$

Simplify:

$$\int{\left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)d x} = \frac{x \left(- 24 x^{4} + 50 x^{2} - 75 x - 60\right)}{30}$$

Add the constant of integration:

$$\int{\left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)d x} = \frac{x \left(- 24 x^{4} + 50 x^{2} - 75 x - 60\right)}{30}+C$$

$$$\int \left(- 4 x^{4} + 5 x^{2} - 5 x - 2\right)\, dx = \frac{x \left(- 24 x^{4} + 50 x^{2} - 75 x - 60\right)}{30} + C$$$A