Integral of $$$\frac{i x}{e^{\frac{3}{2}}}$$$

Find $$$\int \frac{i x}{e^{\frac{3}{2}}}\, dx$$$.

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

$${\color{red}{\int{\frac{i x}{e^{\frac{3}{2}}} d x}}} = {\color{red}{\frac{i \int{x d x}}{e^{\frac{3}{2}}}}}$$

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

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

Therefore,

$$\int{\frac{i x}{e^{\frac{3}{2}}} d x} = \frac{i x^{2}}{2 e^{\frac{3}{2}}}$$

Add the constant of integration:

$$\int{\frac{i x}{e^{\frac{3}{2}}} d x} = \frac{i x^{2}}{2 e^{\frac{3}{2}}}+C$$

$$$\int \frac{i x}{e^{\frac{3}{2}}}\, dx = \frac{i x^{2}}{2 e^{\frac{3}{2}}} + C$$$A