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solve this using u subsististuina with reasoning 

To solve the integral \(\int \frac{x^3}{\sqrt{1 - x^4}} \, dx\) using \(u\)-substitution, we can follow these steps:

1. **Identify the substitution**:
   Let's choose \(u\) such that the integrand simplifies significantly. A good candidate is \(u = 1 - x^4\), since its derivative, when multiplied by \(x^3\), will help us simplify the integral.

2. **Compute the derivative**:
   We need to find \(du\) in terms of \(dx\).
   \[
   u = 1 - x^4 \implies \frac{du}{dx} = -4x^3 \implies du = -4x^3 \, dx
   \]
   Solving for \(dx\), we get:
   \[
   dx = \frac{du}{-4x^3}
   \]

3. **Substitute \(u\) and \(dx\) in the integral**:
   Substitute \(u\) and \(dx\) into the integral:
   \[
   \int \frac{x^3}{\sqrt{1 - x^4}} \, dx = \int \frac{x^3}{\sqrt{u}} \cdot \frac{du}{-4x^3}
   \]

4. **Simplify the integral**:
   \[
   \int \frac{x^3}{\sqrt{u}} \cdot \frac{du}{-4x^3} = \int \frac{1}{-4\sqrt{u}} \, du = -\frac{1}{4} \int u^{-\frac{1}{2}} \, du
   \]

5. **Integrate with respect to \(u\)**:
   \[
   -\frac{1}{4} \int u^{-\frac{1}{2}} \, du = -\frac{1}{4} \cdot 2u^{\frac{1}{2}} + C = -\frac{1}{2} u^{\frac{1}{2}} + C
   \]

6. **Substitute back \(u = 1 - x^4\)**:
   \[
   -\frac{1}{2} u^{\frac{1}{2}} + C = -\frac{1}{2} \sqrt{1 - x^4} + C
   \]

Therefore, the solution to the integral \(\int \frac{x^3}{\sqrt{1 - x^4}} \, dx\) is:
\[
-\frac{1}{2} \sqrt{1 - x^4} + C
\]

Would you like any further details or have any questions?

Here are 8 related questions to further your understanding:

1. How would the integral change if the integrand was \(\int \frac{x^2}{\sqrt{1 - x^4}} \, dx\)?
2. Can you use a different substitution to solve the same integral?
3. What if the integral was \(\int \frac{x^3}{\sqrt{1 + x^4}} \, dx\)?
4. How do you determine the best substitution for an integral?
5. Can this integral be solved using integration by parts?
6. How does the constant of integration \(C\) affect the final solution?
7. What are some common substitutions for integrals involving square roots?
8. How does \(u\)-substitution simplify integrals involving polynomial expressions?

**Tip**: Always double-check your substitution and the corresponding derivative to ensure you are integrating correctly.

Learn how to solve the integral ∫(x^3)/(sqrt(1 - x^4)) dx step-by-step using u-substitution method in calculus. This problem covers advanced integration techniques, including polynomial integrals and trigonometric substitution. Suitable for advanced high school and college students.

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