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The integral provided is:

\[
\int \frac{14 \, dx}{(1 - x^2)^{3/2}}
\]

To solve this, let's use the following approach:

1. **Substitute**: This integral resembles the derivative of the inverse sine function. Recall that:
   \[
   \frac{d}{dx} \left( \arcsin(x) \right) = \frac{1}{\sqrt{1 - x^2}}
   \]

   We can extend this by recognizing a similar integral form where the power of \((1 - x^2)\) is raised to \(\frac{3}{2}\).

2. **Apply a standard form**: We know that:
   \[
   \int \frac{dx}{(1 - x^2)^{3/2}} = \frac{x}{\sqrt{1 - x^2}} + C
   \]

3. **Multiply by the constant**: In this case, we have a constant \(14\) in the numerator, so we multiply our result by \(14\):
   \[
   \int \frac{14 \, dx}{(1 - x^2)^{3/2}} = 14 \cdot \frac{x}{\sqrt{1 - x^2}} + C
   \]

### Final Answer:

\[
\int \frac{14 \, dx}{(1 - x^2)^{3/2}} = \frac{14x}{\sqrt{1 - x^2}} + C
\]

Would you like more details on any step or have any questions?

---

Here are some related questions to deepen your understanding:

1. How would the solution change if the denominator were \((1 + x^2)^{3/2}\) instead of \((1 - x^2)^{3/2}\)?
2. Can we solve this integral using trigonometric substitution?
3. What happens if the exponent in the denominator is different, like \((1 - x^2)^{5/2}\)?
4. How does the inverse trigonometric function property help in evaluating integrals of similar forms?
5. How would we approach this integral if the numerator had a variable, such as \(14x\)?

**Tip:** When dealing with integrals involving terms like \((1 - x^2)^{n}\), checking if it resembles an inverse trigonometric derivative often simplifies the process.

Evaluate the integral: \( \int \frac{14 \, dx}{(1 - x^2)^{3/2}} \)

Learn how to solve the integral \( \int \frac{14 \, dx}{(1 - x^2)^{3/2}} \) using techniques of substitution and properties of inverse trigonometric functions. This problem demonstrates key concepts in integral calculus, suitable for undergraduate calculus students.

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