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The integral to solve is:

\[
\int \frac{1}{\sqrt{7x^2 - 1}} \, dx
\]

### Step-by-Step Solution:

To solve this integral, we will use a trigonometric substitution. Notice that the expression inside the square root is in the form of \(a x^2 - b\), which suggests a substitution involving a hyperbolic function or a trigonometric function.

#### Step 1: Substitution

Consider the substitution:

\[
x = \frac{1}{\sqrt{7}} \sec(\theta)
\]

This implies:

\[
dx = \frac{1}{\sqrt{7}} \sec(\theta) \tan(\theta) \, d\theta
\]

Substitute into the integral:

\[
7x^2 - 1 = 7 \left( \frac{1}{\sqrt{7}} \sec(\theta) \right)^2 - 1 = 7 \frac{1}{7} \sec^2(\theta) - 1 = \sec^2(\theta) - 1 = \tan^2(\theta)
\]

Thus:

\[
\sqrt{7x^2 - 1} = \sqrt{\tan^2(\theta)} = \tan(\theta)
\]

#### Step 2: Rewrite the Integral

Substitute all of this into the integral:

\[
\int \frac{1}{\sqrt{7x^2 - 1}} \, dx = \int \frac{1}{\tan(\theta)} \cdot \frac{1}{\sqrt{7}} \sec(\theta) \tan(\theta) \, d\theta
\]

This simplifies to:

\[
\frac{1}{\sqrt{7}} \int \sec(\theta) \, d\theta
\]

#### Step 3: Integrate

The integral of \(\sec(\theta)\) is:

\[
\int \sec(\theta) \, d\theta = \ln | \sec(\theta) + \tan(\theta) | + C
\]

Thus:

\[
\frac{1}{\sqrt{7}} \ln | \sec(\theta) + \tan(\theta) | + C
\]

#### Step 4: Substitute Back

Recall the original substitution \(x = \frac{1}{\sqrt{7}} \sec(\theta)\), therefore:

\[
\sec(\theta) = \sqrt{7} x, \quad \tan(\theta) = \sqrt{7x^2 - 1}
\]

Substitute back:

\[
\frac{1}{\sqrt{7}} \ln \left| \sqrt{7} x + \sqrt{7x^2 - 1} \right| + C
\]

### Final Answer:

\[
\int \frac{1}{\sqrt{7x^2 - 1}} \, dx = \frac{1}{\sqrt{7}} \ln \left| \sqrt{7} x + \sqrt{7x^2 - 1} \right| + C
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you solve integrals involving expressions like \(\sqrt{ax^2 + b}\)?
2. What is the integral of \(\frac{1}{x^2 - a^2}\)?
3. How do you perform hyperbolic substitutions in integrals?
4. Can you explain the integrals of hyperbolic functions like \(\cosh(x)\) and \(\sinh(x)\)?
5. What is the difference between using trigonometric and hyperbolic substitutions for integrals?

### Tip:

When integrating functions involving \(\sqrt{ax^2 - b}\) or similar forms, trigonometric substitutions are a powerful tool to simplify the integrals!

Learn how to solve the integral ∫ 1 / √(7x^2 - 1) dx using trigonometric substitution. This problem involves advanced calculus concepts and is suitable for advanced high school and college-level students.

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