To evaluate the definite integral of $4x^3 \, dx$, we first find the indefinite integral, then apply the limits if they are given.

Indefinite Integral

The integral we need to evaluate is:

$\int 4x^3 \, dx$

To find the antiderivative, we apply the power rule for integration:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

For $4x^3$, we have:

$\int 4x^3 \, dx = 4 \cdot \int x^3 \, dx = 4 \cdot \left( \frac{x^{3+1}}{3+1} \right) + C = 4 \cdot \frac{x^4}{4} + C = x^4 + C$

So, the indefinite integral is:

$x^4 + C$

Definite Integral (if limits are provided)

If you have specific limits, say from $a$ to $b$, then the definite integral of $4x^3$ from $a$ to $b$ is:

$\int_{a}^{b} 4x^3 \, dx = \left[ x^4 \right]_{a}^{b} = b^4 - a^4$

If you provide the limits, I can calculate the specific value for you.

Would you like to specify the limits for this integral?

Related Questions

1. What is the integral of other simple polynomials, such as $5x^2$?
2. How do you handle integrals of trigonometric functions like $\sin(x)$ or $\cos(x)$?
3. Can you integrate functions involving products, such as $x \cdot e^x$?
4. What are the techniques for integrating rational functions, like $\frac{1}{x}$?
5. How do you calculate improper integrals with infinite limits?

Tip

When integrating polynomials, always apply the power rule: increase the exponent by one and divide by the new exponent.