The integral of \( \sqrt{1 - x^2} \) with respect to \( x \) can be solved using trigonometric substitution. Here's the step-by-step solution:

### Given:
\[
\int \sqrt{1 - x^2} \, dx
\]

### Step 1: Trigonometric Substitution
We use the substitution:
\[
x = \sin \theta, \quad dx = \cos \theta \, d\theta
\]

Now, substitute into the integral:
\[
\sqrt{1 - x^2} = \sqrt{1 - \sin^2 \theta} = \cos \theta
\]

Thus, the integral becomes:
\[
\int \cos^2 \theta \, d\theta
\]

### Step 2: Simplify Using Trig Identity
We can use the identity \( \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \) to simplify the integral:
\[
\int \cos^2 \theta \, d\theta = \int \frac{1 + \cos(2\theta)}{2} \, d\theta
\]

Now, split the integral:
\[
\int \frac{1 + \cos(2\theta)}{2} \, d\theta = \frac{1}{2} \int 1 \, d\theta + \frac{1}{2} \int \cos(2\theta) \, d\theta
\]

### Step 3: Evaluate the Integrals
- \( \frac{1}{2} \int 1 \, d\theta = \frac{\theta}{2} \)
- \( \frac{1}{2} \int \cos(2\theta) \, d\theta = \frac{\sin(2\theta)}{4} \)

Thus, the integral becomes:
\[
\frac{\theta}{2} + \frac{\sin(2\theta)}{4} + C
\]

### Step 4: Back-Substitute \( \theta \)
Recall the substitution \( x = \sin \theta \), so \( \theta = \arcsin(x) \). Also, use the double-angle identity \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) = 2x\sqrt{1 - x^2} \).

Substituting back into the solution:
\[
\frac{\arcsin(x)}{2} + \frac{x \sqrt{1 - x^2}}{2} + C
\]

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

Would you like more details or explanations on any of the steps?

Here are 5 related questions to deepen understanding:
1. How does trigonometric substitution simplify integrals involving square roots?
2. What is the geometric interpretation of the function \( \sqrt{1 - x^2} \)?
3. Can you derive the integral of \( \cos^2 \theta \) using other techniques, such as integration by parts?
4. How would you apply similar techniques to solve \( \int \sqrt{a^2 - x^2} \, dx \)?
5. What other types of integrals are best solved using trigonometric identities?

**Tip:** Trigonometric substitution is especially useful for integrals involving expressions like \( \sqrt{a^2 - x^2} \), as they often simplify into basic trigonometric functions.

Discover how to solve the integral of √(1 - x^2) using trigonometric substitution. This step-by-step guide covers essential calculus techniques, including the use of trigonometric identities and substitution methods. Suitable for undergraduate calculus students.

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