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.