Select all expressions matching the value of indefinite integral int 2^3x-2dx (2^3x-2)/(ln2)+C 3cdot 2^3x-2+C (8^x)/(12ln2)+C

To find the indefinite integral of $\int 2^{3x-2}dx$, we can use the substitution method.<br /><br />Let $u = 3x - 2$, then $du = 3dx$.<br /><br />Substituting $u$ and $du$ into the integral, we get:<br /><br />$\int 2^{3x-2}dx = \int 2^u \cdot \frac{1}{3}du$<br /><br />Now, we can integrate $2^u$ with respect to $u$:<br /><br />$\int 2^u \cdot \frac{1}{3}du = \frac{1}{3} \int 2^u du$<br /><br />Using the formula for the integral of $a^u$, where $a$ is a constant, we have:<br /><br />$\frac{1}{3} \int 2^u du = \frac{1}{3} \cdot \frac{2^u}{\ln 2} + C$<br /><br />Simplifying, we get:<br /><br />$\frac{1}{3} \cdot \frac{2^u}{\ln 2} + C = \frac{2^u}{3 \ln 2} + C$<br /><br />Now, substituting back $u = 3x - 2$, we get:<br /><br />$\frac{2^{3x-2}}{3 \ln 2} + C$<br /><br />Therefore, the correct answer is $\frac{2^{3x-2}}{3 \ln 2} + C$.