Integral of $$$x e^{\frac{1}{2}}$$$

Find $$$\int x e^{\frac{1}{2}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e^{\frac{1}{2}}$$$ and $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{x e^{\frac{1}{2}} d x}}} = {\color{red}{e^{\frac{1}{2}} \int{x d x}}}$$

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

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

Therefore,

$$\int{x e^{\frac{1}{2}} d x} = \frac{x^{2} e^{\frac{1}{2}}}{2}$$

Add the constant of integration:

$$\int{x e^{\frac{1}{2}} d x} = \frac{x^{2} e^{\frac{1}{2}}}{2}+C$$

$$$\int x e^{\frac{1}{2}}\, dx = \frac{x^{2} e^{\frac{1}{2}}}{2} + C$$$A