The image you uploaded contains several integral problems, and they appear to be related to the concept of the indefinite integral and constant of integration. Here's a quick breakdown of each integral expression in the image:

1. $\int \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) \, dx$
2. $\int (4 - 7x)^3 \sqrt{x} \, dx$
3. $\int \frac{dx}{x \sqrt{2x}}$
4. $\int \left( x \sqrt{x} - 5 \right)^2 \, dx$
5. $\int \frac{x^3 - 1}{x - 1} \, dx$

Each of these integrals involves different strategies for integration, such as using power rules, substitution, or simplification. Would you like me to walk through the solution process for any specific integral in this set, or do you have a particular question about these integrals?