Select all expressions matching the value of the definite integral int _(0)^0.54^2xdx (3)/(2ln(4)) 3ln4 (3)/(ln(16)) (3)/(ln(4))

To find the value of the definite integral $\int_{0}^{0.5} 4^{2x} \, dx$, we can use the substitution method.<br /><br />Let $u = 2x$, then $du = 2 \, dx$ and $dx = \frac{du}{2}$.<br /><br />Substituting these values into the integral, we get:<br /><br />$\int_{0}^{0.5} 4^{2x} \, dx = \int_{0}^{1} 4^u \cdot \frac{du}{2} = \frac{1}{2} \int_{0}^{1} 4^u \, du$<br /><br />Now, we can integrate $4^u$ with respect to $u$:<br /><br />$\frac{1}{2} \int_{0}^{1} 4^u \, du = \frac{1}{2} \cdot \frac{4^u}{\ln(4)} \bigg|_{0}^{1} = \frac{1}{2} \cdot \frac{4 - 1}{\ln(4)} = \frac{3}{2 \ln(4)}$<br /><br />Therefore, the correct answer is $\frac{3}{2 \ln(4)}$.