Integral of $$$95 x^{3} - 3 x^{2} - 19 x - 1$$$

Find $$$\int \left(95 x^{3} - 3 x^{2} - 19 x - 1\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(95 x^{3} - 3 x^{2} - 19 x - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} - \int{19 x d x} - \int{3 x^{2} d x} + \int{95 x^{3} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$- \int{19 x d x} - \int{3 x^{2} d x} + \int{95 x^{3} d x} - {\color{red}{\int{1 d x}}} = - \int{19 x d x} - \int{3 x^{2} d x} + \int{95 x^{3} d x} - {\color{red}{x}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=19$$$ and $$$f{\left(x \right)} = x$$$:

$$- x - \int{3 x^{2} d x} + \int{95 x^{3} d x} - {\color{red}{\int{19 x d x}}} = - x - \int{3 x^{2} d x} + \int{95 x^{3} d x} - {\color{red}{\left(19 \int{x d x}\right)}}$$

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

$$- x - \int{3 x^{2} d x} + \int{95 x^{3} d x} - 19 {\color{red}{\int{x d x}}}=- x - \int{3 x^{2} d x} + \int{95 x^{3} d x} - 19 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- x - \int{3 x^{2} d x} + \int{95 x^{3} d x} - 19 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

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

$$- \frac{19 x^{2}}{2} - x + \int{95 x^{3} d x} - {\color{red}{\int{3 x^{2} d x}}} = - \frac{19 x^{2}}{2} - x + \int{95 x^{3} d x} - {\color{red}{\left(3 \int{x^{2} d x}\right)}}$$

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

$$- \frac{19 x^{2}}{2} - x + \int{95 x^{3} d x} - 3 {\color{red}{\int{x^{2} d x}}}=- \frac{19 x^{2}}{2} - x + \int{95 x^{3} d x} - 3 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \frac{19 x^{2}}{2} - x + \int{95 x^{3} d x} - 3 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=95$$$ and $$$f{\left(x \right)} = x^{3}$$$:

$$- x^{3} - \frac{19 x^{2}}{2} - x + {\color{red}{\int{95 x^{3} d x}}} = - x^{3} - \frac{19 x^{2}}{2} - x + {\color{red}{\left(95 \int{x^{3} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=3$$$:

$$- x^{3} - \frac{19 x^{2}}{2} - x + 95 {\color{red}{\int{x^{3} d x}}}=- x^{3} - \frac{19 x^{2}}{2} - x + 95 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- x^{3} - \frac{19 x^{2}}{2} - x + 95 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Therefore,

$$\int{\left(95 x^{3} - 3 x^{2} - 19 x - 1\right)d x} = \frac{95 x^{4}}{4} - x^{3} - \frac{19 x^{2}}{2} - x$$

Simplify:

$$\int{\left(95 x^{3} - 3 x^{2} - 19 x - 1\right)d x} = \frac{x \left(95 x^{3} - 4 x^{2} - 38 x - 4\right)}{4}$$

Add the constant of integration:

$$\int{\left(95 x^{3} - 3 x^{2} - 19 x - 1\right)d x} = \frac{x \left(95 x^{3} - 4 x^{2} - 38 x - 4\right)}{4}+C$$

$$$\int \left(95 x^{3} - 3 x^{2} - 19 x - 1\right)\, dx = \frac{x \left(95 x^{3} - 4 x^{2} - 38 x - 4\right)}{4} + C$$$A