Find the following integral:
$\int \left(x^{\frac{3}{2}} + 2e^{x} - \frac{1}{x}\right) dx$

We use the linearity property of integration to split the integral:
$\int \left(x^{\frac{3}{2}} + 2e^{x} - \frac{1}{x}\right) dx = \int x^{\frac{3}{2}} dx + 2 \int e^{x} dx - \int \frac{1}{x} dx$
Applying the power rule $\int x^{n} dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$,the exponential rule $\int e^{x} dx = e^{x} + C$,and the logarithmic rule $\int \frac{1}{x} dx = \log |x| + C$:
$= \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} + 2e^{x} - \log |x| + C$
$= \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + 2e^{x} - \log |x| + C$
$= \frac{2}{5} x^{\frac{5}{2}} + 2e^{x} - \log |x| + C$