Select all expressions matching the value of the definite integral int _(1)^3(dx)/(3x-1) (ln(4))/(3) (ln(8)-ln(2))/(3) 3ln4

To find the value of the definite integral $\int_{1}^{3} \frac{dx}{3x-1}$, we can use the substitution method.<br /><br />Let's substitute $u = 3x - 1$, then $du = 3dx$, and $dx = \frac{du}{3}$.<br /><br />Now, we can rewrite the integral as:<br /><br />$\int_{1}^{3} \frac{dx}{3x-1} = \int_{2}^{8} \frac{\frac{du}{3}}{u} = \frac{1}{3} \int_{2}^{8} \frac{du}{u} = \frac{1}{3} \ln|u| \bigg|_{2}^{8} = \frac{1}{3} (\ln(8) - \ln(2)) = \frac{\ln(8) - \ln(2)}{3}$<br /><br />Therefore, the correct answer is $\frac{\ln(8) - \ln(2)}{3}$.