Correct Answer:
The correct option is: $e^x\sqrt{x}+C$ (which is shown in the fourth option image).

Step-by-Step Explanation:

1. Identify the given integral:
From the main question image, we are asked to evaluate the following indefinite integral:
$$\int e^x\left(\frac{2x+1}{2\sqrt{x}}\right)dx$$

2. Simplify the integrand fraction:
Let us split the fraction inside the parentheses into two separate terms:
$$\frac{2x+1}{2\sqrt{x}}=\frac{2x}{2\sqrt{x}}+\frac{1}{2\sqrt{x}}$$
Simplifying the first term:
$$\frac{2x}{2\sqrt{x}}=\frac{x}{\sqrt{x}}=\sqrt{x}$$
So, the full expression in the bracket becomes:
$$\sqrt{x}+\frac{1}{2\sqrt{x}}$$

3. Rewrite the integral:
Substituting the simplified expression back into the integral, we get:
$$\int e^x(\sqrt{x}+\frac{1}{2\sqrt{x}})dx$$

4. Apply the standard integral theorem:
Recall the integration formula:
$$\int e^x\left(f\right(x)+f^{\prime }(x\left)\right)dx=e^{x}f\left(x\right)+C$$
Let us choose:
$$f\left(x\right)=\sqrt{x}$$
Now, find the derivative of $f\left(x\right)$ with respect to $x$:
$$f^{\prime }\left(x\right)=\frac{d}{dx}\left(x^{1/2}\right)=\frac{1}{2}{x}^{-1/2}=\frac{1}{2\sqrt{x}}$$

5. Evaluate the integral:
Since our integrand is precisely in the form $e^x\left(f\right(x)+f^{\prime }(x\left)\right)$, we can apply the formula directly:
$$\int e^x(\sqrt{x}+\frac{1}{2\sqrt{x}})dx=e^x\sqrt{x}+C$$
where $C$ is the constant of integration.