To solve this integral, we can use the power rule for antiderivatives. This states that:
∫x^n dx = x^(n+1) / (n+1) + C, where C is the constant of integration.
The power rule allows us to find the antiderivative (the indefinite integral) of x^8, which is x^9 / 9.
Now, we have to account for the limits of the integral, which are 3 and 12. This is done by substituting these values into the antiderivative:
First plug in the upper limit:
12^9 / 9 = 564697676064 / 9.
Then plug in the lower limit:
3^9 / 9 = 19683 / 9.
To get the final result, the value from the lower limit is subtracted from the value from the upper limit, due to the properties of definite integrals:
(564697676064 / 9) - (19683 / 9) = 573306741.
Therefore, the value of the integral from 3 to 12 of x^8 dx is 573306741.