Thanks for bringing the image with the integral problems! I can certainly walk you through the solution process for each of them, but let me first give you an overview of the strategies and how we might approach them:

1) $\int \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) \, dx$

This integral can be handled using the power rule. We express $\sqrt{x}$ as $x^{1/2}$ and $\frac{1}{\sqrt{x}}$ as $x^{-1/2}$, then apply the power rule for each term separately.

2) $\int (4 - 7x)^3 \sqrt{x} \, dx$

For this, a substitution technique could be helpful. Let $u = 4 - 7x$, so the differential of $u$ will simplify the integral. Then, we'd handle the $\sqrt{x}$ term with another substitution.

3) $\int \frac{dx}{x \sqrt{2x}}$

This integral involves simplifying the expression first. Since $\sqrt{2x}$ can be rewritten as $\sqrt{2} \cdot \sqrt{x}$, we might simplify it to an easier expression to integrate, possibly involving a simple substitution for $x$.

4) $\int \left( x \sqrt{x} - 5 \right)^2 \, dx$

For this one, expanding the squared term might be the easiest way to simplify it, and then applying the power rule for each term in the expansion.

5) $\int \frac{x^3 - 1}{x - 1} \, dx$

This is a rational function, and it suggests that polynomial division might simplify it into a more manageable form for integration.

Let me know if you want me to go into more detail on any particular one! Would you like me to solve one of them now, or do you have a specific question about the methods?