To find the antiderivative (also known as the indefinite integral) of the function $4x^2 + 2x - 5$, we integrate each term individually:

$\int (4x^2 + 2x - 5) \, dx$

1. Integrate $4x^2$: $\int 4x^2 \, dx = 4 \cdot \frac{x^{3}}{3} = \frac{4x^3}{3}$

2. Integrate $2x$: $\int 2x \, dx = 2 \cdot \frac{x^{2}}{2} = x^2$

3. Integrate $-5$: $\int -5 \, dx = -5x$

Now, combine the results:

$\frac{4x^3}{3} + x^2 - 5x + C$

Where $C$ is the constant of integration.

Final Answer:

$\int (4x^2 + 2x - 5) \, dx = \frac{4x^3}{3} + x^2 - 5x + C$

Would you like to see more details or have any other questions?

Related Questions:

1. What is the antiderivative of $4x^3 - x^2 + 6$?
2. How do you find the antiderivative of a trigonometric function like $\sin(x)$?
3. Can you explain the process of finding a definite integral for a polynomial?
4. What are some common mistakes when calculating antiderivatives?
5. How do you determine the constant of integration $C$ in specific cases?

Tip:

When integrating polynomials, remember to increase the exponent by 1 and divide by the new exponent for each term.