Integral of $$$x^{5} + 4 x^{4} + 1$$$

Find $$$\int \left(x^{5} + 4 x^{4} + 1\right)\, dx$$$.

Integrate term by term:

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

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

$$\int{4 x^{4} d x} + \int{x^{5} d x} + {\color{red}{\int{1 d x}}} = \int{4 x^{4} d x} + \int{x^{5} d x} + {\color{red}{x}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=5$$$:

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

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

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

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

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

Therefore,

$$\int{\left(x^{5} + 4 x^{4} + 1\right)d x} = \frac{x^{6}}{6} + \frac{4 x^{5}}{5} + x$$

Add the constant of integration:

$$\int{\left(x^{5} + 4 x^{4} + 1\right)d x} = \frac{x^{6}}{6} + \frac{4 x^{5}}{5} + x+C$$

$$$\int \left(x^{5} + 4 x^{4} + 1\right)\, dx = \left(\frac{x^{6}}{6} + \frac{4 x^{5}}{5} + x\right) + C$$$A