To evaluate the integral \(\int \sqrt{x^2 - 1} \, dx\), we can use trigonometric substitution. Here are the steps:

1. **Substitute**: Let \(x = \sec(\theta)\), hence \(dx = \sec(\theta) \tan(\theta) \, d\theta\).

2. **Simplify the Integrand**:
   \[
   \sqrt{x^2 - 1} = \sqrt{\sec^2(\theta) - 1} = \sqrt{\tan^2(\theta)} = |\tan(\theta)|
   \]
   Since \(\sec(\theta) \geq 1\), \(\theta\) ranges from \(0\) to \(\pi/2\) or from \(\pi\) to \(3\pi/2\), and \(\tan(\theta) \geq 0\), we can write \(|\tan(\theta)| = \tan(\theta)\).

3. **Rewrite the Integral**:
   \[
   \int \sqrt{x^2 - 1} \, dx = \int \tan(\theta) \sec(\theta) \tan(\theta) \, d\theta = \int \tan^2(\theta) \sec(\theta) \, d\theta
   \]

4. **Simplify**: Use the identity \(\tan^2(\theta) = \sec^2(\theta) - 1\):
   \[
   \int \tan^2(\theta) \sec(\theta) \, d\theta = \int (\sec^2(\theta) - 1) \sec(\theta) \, d\theta = \int (\sec^3(\theta) - \sec(\theta)) \, d\theta
   \]

5. **Integrate**: The integral can be split into two parts:
   \[
   \int \sec^3(\theta) \, d\theta - \int \sec(\theta) \, d\theta
   \]

   The integrals are:
   \[
   \int \sec(\theta) \, d\theta = \ln |\sec(\theta) + \tan(\theta)| + C_1
   \]
   \[
   \int \sec^3(\theta) \, d\theta = \frac{1}{2} \sec(\theta) \tan(\theta) + \frac{1}{2} \ln |\sec(\theta) + \tan(\theta)| + C_2
   \]

   Combining these:
   \[
   \int \sec^3(\theta) \, d\theta - \int \sec(\theta) \, d\theta = \frac{1}{2} \sec(\theta) \tan(\theta) + \frac{1}{2} \ln |\sec(\theta) + \tan(\theta)| - \ln |\sec(\theta) + \tan(\theta)| + C
   \]
   \[
   = \frac{1}{2} \sec(\theta) \tan(\theta) - \frac{1}{2} \ln |\sec(\theta) + \tan(\theta)| + C
   \]

6. **Substitute Back**: Recall that \(x = \sec(\theta)\), so \(\sec(\theta) = x\) and \(\tan(\theta) = \sqrt{x^2 - 1}\):
   \[
   \int \sqrt{x^2 - 1} \, dx = \frac{1}{2} x \sqrt{x^2 - 1} - \frac{1}{2} \ln |x + \sqrt{x^2 - 1}| + C
   \]

Therefore, the final result is:
\[
\int \sqrt{x^2 - 1} \, dx = \frac{1}{2} x \sqrt{x^2 - 1} - \frac{1}{2} \ln |x + \sqrt{x^2 - 1}| + C
\]

Would you like more details or have any questions?

Here are some questions you might want to explore next:
1. How do you find the integral of \(e^x \sin(x)\)?
2. What are the steps to solve a second-order differential equation?
3. How do you evaluate the limit of a function as \(x\) approaches infinity?
4. What is the series expansion for \(\sin(x)\)?
5. How do you find the derivative of a product of two functions?

**Tip:** For integrals involving square roots of quadratic expressions, trigonometric substitutions are often very useful.

Learn how to evaluate the integral ∫ sqrt(x^2 - 1) dx using trigonometric substitution with a step-by-step solution. This problem involves integral calculus and trigonometric substitution, suitable for advanced high school or college-level 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