Integral of $$$\frac{x^{5}}{15} - \frac{x^{3}}{3} + x$$$

Find $$$\int \left(\frac{x^{5}}{15} - \frac{x^{3}}{3} + x\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(\frac{x^{5}}{15} - \frac{x^{3}}{3} + x\right)d x}}} = {\color{red}{\left(\int{x d x} - \int{\frac{x^{3}}{3} d x} + \int{\frac{x^{5}}{15} d x}\right)}}$$

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

$$- \int{\frac{x^{3}}{3} d x} + \int{\frac{x^{5}}{15} d x} + {\color{red}{\int{x d x}}}=- \int{\frac{x^{3}}{3} d x} + \int{\frac{x^{5}}{15} d x} + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \int{\frac{x^{3}}{3} d x} + \int{\frac{x^{5}}{15} d x} + {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(x \right)} = x^{3}$$$:

$$\frac{x^{2}}{2} + \int{\frac{x^{5}}{15} d x} - {\color{red}{\int{\frac{x^{3}}{3} d x}}} = \frac{x^{2}}{2} + \int{\frac{x^{5}}{15} d x} - {\color{red}{\left(\frac{\int{x^{3} d x}}{3}\right)}}$$

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

$$\frac{x^{2}}{2} + \int{\frac{x^{5}}{15} d x} - \frac{{\color{red}{\int{x^{3} d x}}}}{3}=\frac{x^{2}}{2} + \int{\frac{x^{5}}{15} d x} - \frac{{\color{red}{\frac{x^{1 + 3}}{1 + 3}}}}{3}=\frac{x^{2}}{2} + \int{\frac{x^{5}}{15} d x} - \frac{{\color{red}{\left(\frac{x^{4}}{4}\right)}}}{3}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{15}$$$ and $$$f{\left(x \right)} = x^{5}$$$:

$$- \frac{x^{4}}{12} + \frac{x^{2}}{2} + {\color{red}{\int{\frac{x^{5}}{15} d x}}} = - \frac{x^{4}}{12} + \frac{x^{2}}{2} + {\color{red}{\left(\frac{\int{x^{5} d x}}{15}\right)}}$$

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

$$- \frac{x^{4}}{12} + \frac{x^{2}}{2} + \frac{{\color{red}{\int{x^{5} d x}}}}{15}=- \frac{x^{4}}{12} + \frac{x^{2}}{2} + \frac{{\color{red}{\frac{x^{1 + 5}}{1 + 5}}}}{15}=- \frac{x^{4}}{12} + \frac{x^{2}}{2} + \frac{{\color{red}{\left(\frac{x^{6}}{6}\right)}}}{15}$$

Therefore,

$$\int{\left(\frac{x^{5}}{15} - \frac{x^{3}}{3} + x\right)d x} = \frac{x^{6}}{90} - \frac{x^{4}}{12} + \frac{x^{2}}{2}$$

Add the constant of integration:

$$\int{\left(\frac{x^{5}}{15} - \frac{x^{3}}{3} + x\right)d x} = \frac{x^{6}}{90} - \frac{x^{4}}{12} + \frac{x^{2}}{2}+C$$

$$$\int \left(\frac{x^{5}}{15} - \frac{x^{3}}{3} + x\right)\, dx = \left(\frac{x^{6}}{90} - \frac{x^{4}}{12} + \frac{x^{2}}{2}\right) + C$$$A