How do you integrate the function (4x^3 - 2) with respect to x?

To integrate the function (4x^3 - 2) with respect to (x), we apply the power rule of integration. This rule states that when integrating a polynomial term (x^n), the integral is given by (\frac{x^{n+1}}{n+1}), plus a constant of integration (C).

Breaking down the given function (4x^3 - 2):

1. For the term (4x^3):

• We find the integral using the power rule:

[

\int 4x^3 , dx = 4 \cdot \frac{x^{3+1}}{3+1} = 4 \cdot \frac{x^4}{4} = x^4.

]

2. For the constant term (-2):

• The integral of a constant (a) with respect to (x) is given by:

[

\int a , dx = ax.

]

Thus,

[

\int -2 , dx = -2x.

]

Combining the results from both parts gives:

[

\