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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!

Learn how to solve the integral ∫(dx / sqrt(7x - 1)) with a detailed step-by-step solution using the substitution method. This problem involves integrating a square root function and demonstrates the application of substitution in calculus.

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