Integral of $$$2 \sqrt{x^{5}}$$$

Find $$$\int 2 \sqrt{x^{5}}\, dx$$$.

The input is rewritten: $$$\int{2 \sqrt{x^{5}} d x}=\int{2 x^{\frac{5}{2}} d x}$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = x^{\frac{5}{2}}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{5}{2}$$$:

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

Therefore,

$$\int{2 x^{\frac{5}{2}} d x} = \frac{4 x^{\frac{7}{2}}}{7}$$

Add the constant of integration:

$$\int{2 x^{\frac{5}{2}} d x} = \frac{4 x^{\frac{7}{2}}}{7}+C$$

$$$\int 2 \sqrt{x^{5}}\, dx = \frac{4 x^{\frac{7}{2}}}{7} + C$$$A