Integral of $$$1 - 49 x^{20}$$$

Find $$$\int \left(1 - 49 x^{20}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(1 - 49 x^{20}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{49 x^{20} d x}\right)}}$$

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

$$- \int{49 x^{20} d x} + {\color{red}{\int{1 d x}}} = - \int{49 x^{20} d x} + {\color{red}{x}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=49$$$ and $$$f{\left(x \right)} = x^{20}$$$:

$$x - {\color{red}{\int{49 x^{20} d x}}} = x - {\color{red}{\left(49 \int{x^{20} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=20$$$:

$$x - 49 {\color{red}{\int{x^{20} d x}}}=x - 49 {\color{red}{\frac{x^{1 + 20}}{1 + 20}}}=x - 49 {\color{red}{\left(\frac{x^{21}}{21}\right)}}$$

Therefore,

$$\int{\left(1 - 49 x^{20}\right)d x} = - \frac{7 x^{21}}{3} + x$$

Add the constant of integration:

$$\int{\left(1 - 49 x^{20}\right)d x} = - \frac{7 x^{21}}{3} + x+C$$

$$$\int \left(1 - 49 x^{20}\right)\, dx = \left(- \frac{7 x^{21}}{3} + x\right) + C$$$A