The integral to solve is:

$\int \frac{dx}{\sqrt{7x - 1}}$

To solve this integral, we can use a simple substitution.

Step-by-Step Solution:

1. Substitution:

   Let: $u = 7x - 1$ Then, the derivative of $u$ with respect to $x$ is: $\frac{du}{dx} = 7 \implies dx = \frac{du}{7}$

2. Rewrite the Integral:

   Substituting these into the integral gives: $\int \frac{dx}{\sqrt{7x - 1}} = \int \frac{\frac{du}{7}}{\sqrt{u}}$

   Simplify the expression: $\int \frac{du}{7 \sqrt{u}} = \frac{1}{7} \int u^{-\frac{1}{2}} \, du$

3. Integrate:

   The integral of $u^{-\frac{1}{2}}$ is: $\int u^{-\frac{1}{2}} \, du = 2u^{\frac{1}{2}} = 2\sqrt{u}$

   Therefore: $\frac{1}{7} \int u^{-\frac{1}{2}} \, du = \frac{1}{7} \cdot 2\sqrt{u} = \frac{2\sqrt{u}}{7}$

4. Substitute Back:

   Replace $u$ with $7x - 1$: $\frac{2\sqrt{7x - 1}}{7} + C$

   Where $C$ is the constant of integration.

Final Answer:

$\int \frac{dx}{\sqrt{7x - 1}} = \frac{2\sqrt{7x - 1}}{7} + C$

Would you like more details or have any questions?

Related Questions:

1. How would you solve an integral involving a different linear expression inside the square root?
2. What are the steps for solving an integral of the form $\int \frac{dx}{ax + b}$?
3. How do you perform integration using trigonometric substitution?
4. What are the different techniques of integration?
5. How would you integrate functions involving higher powers of expressions, such as $(ax + b)^n$?

Tip:

When faced with integrals involving square roots of linear expressions, always consider substitution to simplify the integral!