solve integral from negative infinity to infinity of x^3/(x^4+1)

We are asked to solve the integral:

\[
I = \int_{-\infty}^{\infty} \frac{x^3}{x^4 + 1} \, dx
\]

### **Solution:**

Since the integrand is an odd function (i.e., \( f(-x) = -f(x) \)), the integral over the symmetric interval \((-\infty, \infty)\) will be **zero**. Here is the reasoning in detail:

#### **Step 1: Check for symmetry**  
The integrand is:

\[
f(x) = \frac{x^3}{x^4 + 1}
\]

Now, substitute \( x \rightarrow -x \) to test for symmetry:

\[
f(-x) = \frac{(-x)^3}{(-x)^4 + 1} = -\frac{x^3}{x^4 + 1} = -f(x)
\]

This shows that the integrand is **odd**.

#### **Step 2: Integral of an odd function over symmetric limits**  
For any odd function \( f(x) \), the integral over symmetric limits (such as from \(-\infty\) to \(\infty\)) is zero:

\[
\int_{-\infty}^{\infty} f(x) \, dx = 0
\]

Thus, the result is:

\[
\boxed{0}
\]

---

### Conclusion  
The integral of \( \frac{x^3}{x^4 + 1} \) from \(-\infty\) to \( \infty \) is \( 0 \) due to the odd nature of the integrand.

---

Do you have any questions or need further details?

### Related Questions:
1. How do you test whether a function is odd or even?
2. What are the properties of integrals of odd functions over symmetric intervals?
3. Can the integral of an even function from \(-\infty\) to \(\infty\) ever be zero?
4. How would the result change if the limits were \(0\) to \(\infty\) instead?
5. What are the typical methods to solve integrals involving rational functions?

#### **Tip:**  
When facing symmetric integrals, always check whether the integrand is odd or even—it can simplify the problem immensely.

Learn how to solve the definite integral of x^3/(x^4 + 1) from negative infinity to infinity. This problem explores key calculus concepts such as odd functions, symmetry in integrals, and properties of definite integrals. Suitable for undergraduate students in Calculus I/II.

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