Integral of $$$- \frac{x^{26}}{2} + 3 x - 1$$$

Find $$$\int \left(- \frac{x^{26}}{2} + 3 x - 1\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- \frac{x^{26}}{2} + 3 x - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} + \int{3 x d x} - \int{\frac{x^{26}}{2} d x}\right)}}$$

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

$$\int{3 x d x} - \int{\frac{x^{26}}{2} d x} - {\color{red}{\int{1 d x}}} = \int{3 x d x} - \int{\frac{x^{26}}{2} d x} - {\color{red}{x}}$$

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

$$- x - \int{\frac{x^{26}}{2} d x} + {\color{red}{\int{3 x d x}}} = - x - \int{\frac{x^{26}}{2} d x} + {\color{red}{\left(3 \int{x d x}\right)}}$$

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

$$- x - \int{\frac{x^{26}}{2} d x} + 3 {\color{red}{\int{x d x}}}=- x - \int{\frac{x^{26}}{2} d x} + 3 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- x - \int{\frac{x^{26}}{2} d x} + 3 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = x^{26}$$$:

$$\frac{3 x^{2}}{2} - x - {\color{red}{\int{\frac{x^{26}}{2} d x}}} = \frac{3 x^{2}}{2} - x - {\color{red}{\left(\frac{\int{x^{26} d x}}{2}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=26$$$:

$$\frac{3 x^{2}}{2} - x - \frac{{\color{red}{\int{x^{26} d x}}}}{2}=\frac{3 x^{2}}{2} - x - \frac{{\color{red}{\frac{x^{1 + 26}}{1 + 26}}}}{2}=\frac{3 x^{2}}{2} - x - \frac{{\color{red}{\left(\frac{x^{27}}{27}\right)}}}{2}$$

Therefore,

$$\int{\left(- \frac{x^{26}}{2} + 3 x - 1\right)d x} = - \frac{x^{27}}{54} + \frac{3 x^{2}}{2} - x$$

Simplify:

$$\int{\left(- \frac{x^{26}}{2} + 3 x - 1\right)d x} = \frac{x \left(- x^{26} + 81 x - 54\right)}{54}$$

Add the constant of integration:

$$\int{\left(- \frac{x^{26}}{2} + 3 x - 1\right)d x} = \frac{x \left(- x^{26} + 81 x - 54\right)}{54}+C$$

$$$\int \left(- \frac{x^{26}}{2} + 3 x - 1\right)\, dx = \frac{x \left(- x^{26} + 81 x - 54\right)}{54} + C$$$A