To solve the integral of \( \frac{3}{x\sqrt{x^2 - 1}} \), we can use a trigonometric substitution. Here's the step-by-step process:

### Step 1: Substitution
We can use the substitution:
\[
x = \sec(\theta)
\]
Then, \( dx = \sec(\theta)\tan(\theta) d\theta \), and the expression \( \sqrt{x^2 - 1} \) simplifies to \( \sqrt{\sec^2(\theta) - 1} = \tan(\theta) \).

### Step 2: Substituting into the Integral
Substitute \( x = \sec(\theta) \) and \( dx = \sec(\theta)\tan(\theta) d\theta \) into the integral:

\[
\int \frac{3}{x\sqrt{x^2 - 1}} dx = \int \frac{3}{\sec(\theta)\tan(\theta)} \cdot \sec(\theta)\tan(\theta) d\theta
\]

This simplifies to:
\[
\int 3 d\theta
\]

### Step 3: Solve the Integral
The integral of \( 3 \, d\theta \) is straightforward:
\[
3\theta + C
\]

### Step 4: Back-Substitute \( \theta \)
Since \( x = \sec(\theta) \), we have \( \theta = \sec^{-1}(x) \). Therefore, the solution is:

\[
3\sec^{-1}(x) + C
\]

### Final Answer:
\[
\int \frac{3}{x\sqrt{x^2 - 1}} dx = 3\sec^{-1}(x) + C
\]

Would you like more details on trigonometric substitution or any other step?

---

Here are 5 related questions you might find useful:
1. What are other integrals that involve trigonometric substitution?
2. How does one choose the correct substitution for integrals involving square roots?
3. What is the derivative of \( \sec^{-1}(x) \)?
4. How can you apply hyperbolic substitution for similar integrals?
5. How would you integrate \( \frac{1}{x^2\sqrt{x^2 - 1}} \)?

**Tip:** Trigonometric substitution is especially useful for integrals involving expressions like \( \sqrt{x^2 - 1} \), \( \sqrt{x^2 + 1} \), or \( \sqrt{1 - x^2} \).

Learn how to solve the integral ∫(3 / x sqrt(x^2 - 1)) dx using trigonometric substitution. This step-by-step guide covers key calculus concepts such as trigonometric substitution, integral simplification, and back-substitution. Suitable for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation