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 (2^3x-2)/(3ln2)+C

The correct answer is $\frac {2^{3x-2}}{ln2}+C$.<br /><br />To evaluate the indefinite integral $\int 2^{3x-2}dx$, we can use the substitution method. Let $u = 3x - 2$, then $du = 3dx$, and $dx = \frac{du}{3}$.<br /><br />Substituting these values into the integral, we get:<br /><br />$\int 2^{3x-2}dx = \int 2^u \cdot \frac{du}{3} = \frac{1}{3} \int 2^u du$<br /><br />Now, we can integrate $2^u$ with respect to $u$:<br /><br />$\frac{1}{3} \int 2^u du = \frac{1}{3} \cdot \frac{2^u}{\ln 2} + C = \frac{2^u}{3 \ln 2} + C$<br /><br />Finally, substituting back $u = 3x - 2$, we get:<br /><br />$\frac{2^{3x-2}}{3 \ln 2} + C$<br /><br />Therefore, the correct expression matching the value of the indefinite integral is $\frac {2^{3x-2}}{ln2}+C$.