The integral presented is:

$\int_{-1}^{2} x^4 \, dx$

Step 1: Solve the indefinite integral

The integral of $x^4$ is calculated as: $\int x^4 \, dx = \frac{x^5}{5} + C$

Step 2: Apply the definite integral limits

We evaluate this from $-1$ to $2$. Using the formula for definite integrals: $\int_{-1}^{2} x^4 \, dx = \left[ \frac{x^5}{5} \right]_{-1}^{2}$

Substituting the upper and lower limits:

1. For $x = 2$: $\frac{2^5}{5} = \frac{32}{5}$

2. For $x = -1$: $\frac{(-1)^5}{5} = \frac{-1}{5}$

Step 3: Compute the result

Subtract the lower limit value from the upper limit value: $\frac{32}{5} - \left(\frac{-1}{5}\right) = \frac{32}{5} + \frac{1}{5} = \frac{33}{5}$

Final Answer:

$\int_{-1}^{2} x^4 \, dx = \frac{33}{5}$

Would you like further details or clarifications?

Here are 5 related questions to expand your understanding:

1. How do you compute definite integrals for higher powers of $x$?
2. Why do constants like $C$ disappear in definite integrals?
3. What are the common methods for approximating definite integrals?
4. Can this integral be solved using numerical methods instead of symbolic computation?
5. How do definite integrals relate to the area under the curve?

Tip: Always simplify powers and constants first to reduce the chance of computational errors when solving integrals!