To find the integral of \( x^3 \) from 0 to 1, we use the fundamental theorem of calculus. Here are the steps:

1. **Find the antiderivative of \( x^3 \)**:
   \[
   \int x^3 \, dx = \frac{x^4}{4} + C
   \]

2. **Evaluate the antiderivative at the bounds 0 and 1**:
   \[
   \left. \frac{x^4}{4} \right|_0^1 = \frac{1^4}{4} - \frac{0^4}{4}
   \]

3. **Simplify the result**:
   \[
   \frac{1}{4} - 0 = \frac{1}{4}
   \]

So, the value of the integral \( \int_0^1 x^3 \, dx \) is \( \frac{1}{4} \).

Would you like any further details or have any questions?

Here are 5 more questions you might consider asking:
1. How do you integrate more complex functions?
2. What are the properties of definite integrals?
3. How do you apply integration by parts?
4. Can you explain the substitution method in integration?
5. How do you solve improper integrals?

**Tip:** When solving definite integrals, always remember to evaluate the antiderivative at the upper and lower bounds and subtract the lower value from the upper value.

Learn how to compute the integral of x^3 from 0 to 1 with a detailed step-by-step solution. This problem involves calculus and the fundamental theorem of calculus, covering antiderivatives and definite integral evaluation. Suitable for advanced high school or college students.

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