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The given integral to evaluate is:

\[
\int x^4 e^{-x^5} \, dx
\]

### Step-by-step Solution:

1. **Substitution Method**: 
   We recognize that the exponent of the exponential function \(-x^5\) suggests a substitution where \( u = -x^5 \).

2. **Differentiate \( u \)**:
   \[
   du = -5x^4 \, dx
   \]
   Hence, 
   \[
   \frac{du}{-5} = x^4 \, dx
   \]

3. **Rewrite the Integral**:
   Substituting into the original integral:
   \[
   \int x^4 e^{-x^5} \, dx = \int e^u \cdot \frac{du}{-5}
   \]

4. **Simplify the Integral**:
   \[
   = -\frac{1}{5} \int e^u \, du
   \]

   The integral of \( e^u \) is just \( e^u \), so we get:
   \[
   -\frac{1}{5} e^u + C
   \]

5. **Substitute Back \( u = -x^5 \)**:
   \[
   -\frac{1}{5} e^{-x^5} + C
   \]

Thus, the solution is:

\[
\boxed{B. \frac{1}{5} e^{-x^5} + C}
\]

Would you like more details or further explanation on this solution?

Here are 5 related questions to deepen your understanding:
1. How do you recognize when to use substitution in integrals?
2. What is the importance of the chain rule in the differentiation process for substitution?
3. How would you evaluate an integral without exponential functions?
4. Can the integral result vary depending on the choice of substitution variable?
5. Why is the constant of integration (\(C\)) essential in indefinite integrals?

**Tip:** Always verify your substitution by differentiating your result to check if it matches the original integrand.

Evaluate the integral: ∫ x^4 e^(-x^5) dx

Learn how to evaluate the integral ∫ x^4 e^(-x^5) dx step-by-step using the substitution method. This calculus problem explores integration techniques commonly used in advanced high school or early college math courses.

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