To solve the integral $\int \sqrt{x^2 - 1} \, dx$, we can use a trigonometric substitution. Here’s a step-by-step solution:

1. Trigonometric Substitution: Let $x = \sec(\theta)$. Then $dx = \sec(\theta)\tan(\theta) \, d\theta$.

   Also, we have: $\sqrt{x^2 - 1} = \sqrt{\sec^2(\theta) - 1} = \sqrt{\tan^2(\theta)} = |\tan(\theta)|$

   Since $\theta$ ranges such that $\sec(\theta) \geq 1$, $\tan(\theta)$ is positive, and $|\tan(\theta)| = \tan(\theta)$.

2. Substitute and Simplify: Substitute $x = \sec(\theta)$ and $dx = \sec(\theta)\tan(\theta) \, d\theta$: $\int \sqrt{x^2 - 1} \, dx = \int \tan(\theta) \cdot \sec(\theta)\tan(\theta) \, d\theta = \int \sec(\theta)\tan^2(\theta) \, d\theta$

3. Simplify the Integral: Use the identity $\tan^2(\theta) = \sec^2(\theta) - 1$: $\int \sec(\theta)\tan^2(\theta) \, d\theta = \int \sec(\theta)(\sec^2(\theta) - 1) \, d\theta$ $= \int \sec^3(\theta) \, d\theta - \int \sec(\theta) \, d\theta$

4. Integrate Separately:

   • The integral $\int \sec(\theta) \, d\theta = \ln|\sec(\theta) + \tan(\theta)| + C$.
   • The integral $\int \sec^3(\theta) \, d\theta$ can be approached using integration by parts or other techniques, but for now, let's use a known result: $\int \sec^3(\theta) \, d\theta = \frac{1}{2} \sec(\theta) \tan(\theta) + \frac{1}{2} \ln|\sec(\theta) + \tan(\theta)| + C$

5. Combine Results: Combine the integrals: $\int \sec^3(\theta) \, d\theta - \int \sec(\theta) \, d\theta = \left(\frac{1}{2} \sec(\theta) \tan(\theta) + \frac{1}{2} \ln|\sec(\theta) + \tan(\theta)|\right) - \ln|\sec(\theta) + \tan(\theta)| + C$ $= \frac{1}{2} \sec(\theta) \tan(\theta) - \frac{1}{2} \ln|\sec(\theta) + \tan(\theta)| + C$

6. Substitute Back: Substitute back $x = \sec(\theta)$ and $\tan(\theta) = \sqrt{x^2 - 1}$: $= \frac{1}{2} x \sqrt{x^2 - 1} - \frac{1}{2} \ln|x + \sqrt{x^2 - 1}| + C$

Therefore, the integral $\int \sqrt{x^2 - 1} \, dx$ 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 five more questions you might be interested in:

1. How do you solve $\int \sin^2(x) \, dx$?
2. What is the derivative of $\ln(x^2 + 1)$?
3. Can you explain the method of partial fractions?
4. How do you find the limit $\lim_{{x \to 0}} \frac{\sin(x)}{x}$?
5. How do you solve a second-order linear differential equation?

Tip: When solving integrals, always check if a trigonometric substitution or another method like integration by parts can simplify the problem.