$$ \int\frac{1}{\sqrt{a^2-x^2}} $$$$ \differentialD x $$

The integral you provided is:

$$
\int \frac{1}{\sqrt{a^2 - x^2}} \, dx
$$

This is a standard integral that can be solved using a trigonometric substitution. The appropriate substitution is:

$$
x = a \sin(\theta)
$$

### Steps to solve:

1. **Substitute** $ x = a \sin(\theta) $, then $ dx = a \cos(\theta) \, d\theta $.

2. The integrand becomes:
   $$
   \int \frac{1}{\sqrt{a^2 - a^2 \sin^2(\theta)}} \cdot a \cos(\theta) \, d\theta
   $$

3. Simplify the square root:
   $$
   \sqrt{a^2 - a^2 \sin^2(\theta)} = \sqrt{a^2 (1 - \sin^2(\theta))} = \sqrt{a^2 \cos^2(\theta)} = a \cos(\theta)
   $$
   So the integrand simplifies to:
   $$
   \int \frac{a \cos(\theta)}{a \cos(\theta)} \, d\theta = \int 1 \, d\theta
   $$

4. Integrate with respect to $ \theta $:
   $$
   \theta + C
   $$

5. Convert back to $ x $ using $ \theta = \arcsin\left(\frac{x}{a}\right) $:
   $$
   \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C
   $$

### Final Answer:
$$
\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C
$$

Would you like any more details on this solution, or do you have any questions?

Here are five related questions you might find interesting:

1. How would you solve the integral $ \int \frac{x}{\sqrt{a^2 - x^2}} \, dx $?
2. What is the integral of $ \int \frac{1}{\sqrt{a^2 + x^2}} \, dx $?
3. How does the substitution $ x = a \sin(\theta) $ work geometrically in this context?
4. What is the significance of the constant $ C $ in indefinite integrals?
5. Can you derive the integral using a different substitution, like $ x = a \cos(\theta) $?

**Tip:** When dealing with integrals involving square roots like this, trigonometric substitutions are often useful because they simplify the radical expression.

Learn how to solve the integral ∫(1 / √(a² - x²)) dx using trigonometric substitution. This detailed step-by-step guide covers key calculus concepts, such as inverse trigonometric functions and the substitution x = a sin(θ). Perfect for students in advanced calculus courses.

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