First, we can split the integral into three separate integrals:
$$\int\left(\sqrt{x}+\frac{3}{x}-2 e^{x}\right) dx = \int \sqrt{x} dx + \int \frac{3}{x} dx - 2 \int e^x dx$$
Now, we can find the indefinite integral for each term:
1) For the first term, $\int
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