To evaluate the given definite integral, we can use the power rule of integration, which states that:
\(\int x^n dx = \frac{x^{n+1}}{n+1} + C\)
where C is the constant of integration.
Using this rule, we can integrate the integrand (3x - 2)^5 as follows:
\(\int (3x - 2)^5 dx = \frac{(3x - 2)^6}{18} + C\)
We have added the constant of integration C to the end of the result since integration is an inverse operation of differentiation and any constant added during differentiation will be lost during integration.
Now, to evaluate the definite integral from 1 to 3, we can substitute the limits of integration into the expression we just found and then subtract the value of the expression at the lower limit from the value at the upper limit, as follows:
\(\int^3_1 (3x - 2)^5 dx = [\frac{(3x - 2)^6}{18}]^3_1\)
\(= \frac{(3(3) - 2)^6}{18} - \frac{(3(1) - 2)^6}{18}\)
\(= \frac{145152}{18}\)
\(= 8064\)
Therefore, the value of the definite integral \(\int^3_1(3x - 2)^5 dx\) is 8064.
