Integral of $$$5 x^{6} - x^{3} + 2 x^{2} + 21$$$

Find $$$\int \left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)d x}}} = {\color{red}{\left(\int{21 d x} + \int{2 x^{2} d x} - \int{x^{3} d x} + \int{5 x^{6} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=21$$$:

$$\int{2 x^{2} d x} - \int{x^{3} d x} + \int{5 x^{6} d x} + {\color{red}{\int{21 d x}}} = \int{2 x^{2} d x} - \int{x^{3} d x} + \int{5 x^{6} d x} + {\color{red}{\left(21 x\right)}}$$

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

$$21 x + \int{2 x^{2} d x} + \int{5 x^{6} d x} - {\color{red}{\int{x^{3} d x}}}=21 x + \int{2 x^{2} d x} + \int{5 x^{6} d x} - {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=21 x + \int{2 x^{2} d x} + \int{5 x^{6} d x} - {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

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

$$- \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + {\color{red}{\int{2 x^{2} d x}}} = - \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + {\color{red}{\left(2 \int{x^{2} d x}\right)}}$$

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

$$- \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + 2 {\color{red}{\int{x^{2} d x}}}=- \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + 2 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + 2 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=5$$$ and $$$f{\left(x \right)} = x^{6}$$$:

$$- \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + {\color{red}{\int{5 x^{6} d x}}} = - \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + {\color{red}{\left(5 \int{x^{6} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=6$$$:

$$- \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + 5 {\color{red}{\int{x^{6} d x}}}=- \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + 5 {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=- \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + 5 {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$

Therefore,

$$\int{\left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)d x} = \frac{5 x^{7}}{7} - \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x$$

Simplify:

$$\int{\left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)d x} = \frac{x \left(60 x^{6} - 21 x^{3} + 56 x^{2} + 1764\right)}{84}$$

Add the constant of integration:

$$\int{\left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)d x} = \frac{x \left(60 x^{6} - 21 x^{3} + 56 x^{2} + 1764\right)}{84}+C$$

$$$\int \left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)\, dx = \frac{x \left(60 x^{6} - 21 x^{3} + 56 x^{2} + 1764\right)}{84} + C$$$A