∫(3x+4)³ dx

The integral shown in the image is:

\int (3x+4) \, dx

To solve this integral, we will use the basic rules of integration.

  1. Integral of 3x:

    • Since the derivative of x^2 is 2x, the integral of x (in terms of x^2) adjusted for the coefficient 3 gives us \frac{3}{2}x^2.
  2. Integral of 4:

    • The integral of a constant is just the constant times the variable of integration, so this would be 4x.

Putting it all together, the integral becomes:
\frac{3}{2}x^2 + 4x + C

where C is the constant of integration.