To integrate the function $\frac{1}{(1-x^2)^{3/2}}$, we can use a substitution method that avoids trigonometric functions. Here's how we can do it step by step:

Step 1: Substitution

Let $u = 1 - x^2$. Then, we find the differential: $du = -2x \, dx \quad \Rightarrow \quad dx = -\frac{du}{2x}$ Also, we express $x$ in terms of $u$: $x^2 = 1 - u \quad \Rightarrow \quad x = \sqrt{1 - u} \text{ (considering the positive branch for simplicity)}$

Step 2: Rewrite the Integral

Substituting $u$ into the integral, we have: $\int \frac{1}{(1-x^2)^{3/2}} \, dx = \int \frac{1}{u^{3/2}} \left(-\frac{du}{2\sqrt{1-u}}\right)$ This gives us: $-\frac{1}{2} \int \frac{1}{u^{3/2} \sqrt{1-u}} \, du$

Step 3: Simplifying the Integral

Now we will simplify: $-\frac{1}{2} \int u^{-3/2} (1-u)^{-1/2} \, du$

Step 4: Beta Function or Standard Integral

The integral $\int u^{-3/2} (1-u)^{-1/2} \, du$ can be related to the Beta function. We recognize that: $\int u^{p-1} (1-u)^{q-1} \, du = B(p, q) = \frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$ For our case, set $p = \frac{1}{2}$ and $q = \frac{1}{2}$: $B\left(-\frac{1}{2}, \frac{1}{2}\right)$ However, we can find this integral explicitly as: $\int u^{-3/2} (1-u)^{-1/2} \, du = -2\sqrt{1-u} + C$

Step 5: Substitute Back

Substituting back to $u = 1 - x^2$: $-\frac{1}{2} \left(-2\sqrt{1-(1-x^2)}\right) = \sqrt{x^2} = |x|$

Final Result

Thus, the integral evaluates to: $\int \frac{1}{(1-x^2)^{3/2}} \, dx = |x| + C$

Summary

So the integral of $\frac{1}{(1-x^2)^{3/2}} \, dx$ is: $|x| + C$

If you need more details or have any questions, feel free to ask! Here are some follow-up questions you might consider:

1. Would you like to see a graphical representation of the integral?
2. Do you want to explore other methods for this integral?
3. Are you interested in the application of this integral in physics or engineering?
4. Would you like to discuss the convergence of similar integrals?
5. Do you want to see how this relates to other functions or transforms?

Tip: Always check your substitutions to ensure they simplify the problem effectively.