https://file.aiquickdraw.com/mathbot/file/1726985684184vm5d2w1j.jpg

The integral given in the image is:

\[
\int \frac{x^2}{e^{x^3} + e^{-x^3}} \, dx
\]

### Step-by-step Solution:

1. **Simplifying the denominator:**
   The denominator \( e^{x^3} + e^{-x^3} \) resembles a form related to hyperbolic functions. In fact:
   
   \[
   e^{x^3} + e^{-x^3} = 2 \cosh(x^3)
   \]
   where \( \cosh(x) \) is the hyperbolic cosine function defined as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).

   So the integral becomes:
   \[
   \int \frac{x^2}{2 \cosh(x^3)} \, dx
   \]

2. **Substitution:**
   Let's make the substitution \( u = x^3 \), which implies \( du = 3x^2 \, dx \), or \( \frac{du}{3} = x^2 \, dx \).

   The integral now transforms into:
   \[
   \int \frac{\frac{du}{3}}{2 \cosh(u)} = \frac{1}{6} \int \frac{du}{\cosh(u)}
   \]

3. **Evaluating the integral:**
   The integral of \( \frac{1}{\cosh(u)} \) is the hyperbolic secant function, which has the following standard result:
   
   \[
   \int \frac{1}{\cosh(u)} \, du = 2 \arctan(\tanh(\frac{u}{2}))
   \]

4. **Substituting back:**
   Since \( u = x^3 \), we get:
   \[
   \frac{1}{6} \cdot 2 \arctan(\tanh(\frac{x^3}{2})) + C = \frac{1}{3} \arctan(\tanh(\frac{x^3}{2})) + C
   \]

Thus, the value of the integral is:

\[
\boxed{\frac{1}{3} \arctan\left(\tanh\left(\frac{x^3}{2}\right)\right) + C}
\]

Let me know if you want more details or if you have any questions!

### Related Questions:
1. What are the properties of hyperbolic functions, like \( \cosh(x) \) and \( \sinh(x) \)?
2. How does substitution work in definite and indefinite integrals?
3. What is the relationship between trigonometric and hyperbolic functions?
4. How is the arctangent function related to the hyperbolic tangent?
5. Can this integral be evaluated numerically for specific values of \( x \)?

### Tip:
Hyperbolic functions often appear in integrals involving exponential terms. They simplify complex expressions and are closely related to trigonometric identities.

Evaluate the integral of (x^2 / (e^(x^3) + e^(-x^3))) dx.

This problem demonstrates how to evaluate the integral of x^2 / (e^(x^3) + e^(-x^3)) using hyperbolic functions and substitution. Learn the step-by-step solution, including how to apply the properties of hyperbolic cosine and the substitution method.

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