Integral of $$$x^{\frac{3}{2}} + 2$$$

Find $$$\int \left(x^{\frac{3}{2}} + 2\right)\, dx$$$.

Integrate term by term:

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

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=2$$$:

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

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

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

Therefore,

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

Add the constant of integration:

$$\int{\left(x^{\frac{3}{2}} + 2\right)d x} = \frac{2 x^{\frac{5}{2}}}{5} + 2 x+C$$

$$$\int \left(x^{\frac{3}{2}} + 2\right)\, dx = \left(\frac{2 x^{\frac{5}{2}}}{5} + 2 x\right) + C$$$A