Integral of $$$3024 \sqrt{14} x^{\frac{3}{2}}$$$

Find $$$\int 3024 \sqrt{14} x^{\frac{3}{2}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3024 \sqrt{14}$$$ and $$$f{\left(x \right)} = x^{\frac{3}{2}}$$$:

$${\color{red}{\int{3024 \sqrt{14} x^{\frac{3}{2}} d x}}} = {\color{red}{\left(3024 \sqrt{14} \int{x^{\frac{3}{2}} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=\frac{3}{2}$$$:

$$3024 \sqrt{14} {\color{red}{\int{x^{\frac{3}{2}} d x}}}=3024 \sqrt{14} {\color{red}{\frac{x^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}=3024 \sqrt{14} {\color{red}{\left(\frac{2 x^{\frac{5}{2}}}{5}\right)}}$$

Therefore,

$$\int{3024 \sqrt{14} x^{\frac{3}{2}} d x} = \frac{6048 \sqrt{14} x^{\frac{5}{2}}}{5}$$

Add the constant of integration:

$$\int{3024 \sqrt{14} x^{\frac{3}{2}} d x} = \frac{6048 \sqrt{14} x^{\frac{5}{2}}}{5}+C$$

$$$\int 3024 \sqrt{14} x^{\frac{3}{2}}\, dx = \frac{6048 \sqrt{14} x^{\frac{5}{2}}}{5} + C$$$A