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To solve the integral \(\int \frac{dx}{(1 - x^2)^{3/2}}\) using trigonometric substitution, we follow these steps:

### Step 1: Trigonometric Substitution
We use the substitution:
\[
x = \sin \theta \quad \text{or} \quad x = \cos \theta
\]
For this integral, we choose \(x = \sin \theta\), hence:
\[
dx = \cos \theta \, d\theta
\]

### Step 2: Rewrite the Integral
Substitute \(x = \sin \theta\) and \(dx = \cos \theta \, d\theta\) into the integral:
\[
\int \frac{dx}{(1 - x^2)^{3/2}} = \int \frac{\cos \theta \, d\theta}{(1 - \sin^2 \theta)^{3/2}}
\]

### Step 3: Simplify the Denominator
Using the identity \(1 - \sin^2 \theta = \cos^2 \theta\), we get:
\[
\int \frac{\cos \theta \, d\theta}{(\cos^2 \theta)^{3/2}} = \int \frac{\cos \theta \, d\theta}{\cos^3 \theta}
\]
Simplify this expression:
\[
\int \frac{1}{\cos^2 \theta} \, d\theta = \int \sec^2 \theta \, d\theta
\]

### Step 4: Integrate
The integral of \(\sec^2 \theta\) is:
\[
\int \sec^2 \theta \, d\theta = \tan \theta + C
\]

### Step 5: Back-Substitution
Recall the substitution \(x = \sin \theta\), so:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{x}{\sqrt{1 - x^2}}
\]

Thus, the final solution is:
\[
\int \frac{dx}{(1 - x^2)^{3/2}} = \frac{x}{\sqrt{1 - x^2}} + C
\]

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. How would you solve the integral \(\int \frac{dx}{(1 - x^2)^{1/2}}\) using trigonometric substitution?
2. Can you find the integral \(\int \frac{x \, dx}{(1 - x^2)^{3/2}}\) using the same substitution?
3. How would you solve \(\int \frac{dx}{(4 - x^2)^{3/2}}\) with trigonometric substitution?
4. What is the result of \(\int \frac{dx}{(1 + x^2)^{3/2}}\) using a different trigonometric substitution?
5. How would the substitution differ for \(\int \frac{dx}{(x^2 - 1)^{3/2}}\)?

**Tip**: Always remember to check if the substitution fits the integral's form to simplify the expression before solving.

Solve the integral \(\int \frac{dx}{(1 - x^2)^{3/2}}\) using trigonometric substitution.

Learn how to solve the integral \(\int \frac{dx}{(1 - x^2)^{3/2}}\) using trigonometric substitution. This solution covers key calculus concepts such as substitution and integration using trigonometric identities. Suitable for students studying college-level calculus.

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