Select all expressions matching the value of indefinite integral int 0.5^3x+5dx (3x+5)0.5^3x+4+C -(2^-3x-5)/(3ln2)+C -(0.5^3x+5)/(ln2)+C (0.5^3x+5)/(3ln0.5)+C

To find the indefinite integral $\int 0.5^{3x+5}dx$, we can use the substitution method.<br /><br />Let $u = 3x + 5$, then $du = 3dx$, and $dx = \frac{du}{3}$.<br /><br />Substituting these values into the integral, we get:<br /><br />$\int 0.5^{3x+5}dx = \int 0.5^u \cdot \frac{du}{3}$<br /><br />Now, we can integrate with respect to $u$:<br /><br />$\int 0.5^u \cdot \frac{du}{3} = \frac{1}{3} \int 0.5^u du$<br /><br />Using the formula for the integral of an exponential function, we have:<br /><br />$\frac{1}{3} \int 0.5^u du = \frac{1}{3} \cdot \frac{0.5^u}{\ln(0.5)} + C$<br /><br />Simplifying further, we get:<br /><br />$\frac{1}{3} \cdot \frac{0.5^u}{\ln(0.5)} + C = \frac{0.5^u}{3 \ln(0.5)} + C$<br /><br />Now, substituting back $u = 3x + 5$, we get:<br /><br />$\frac{0.5^{3x+5}}{3 \ln(0.5)} + C$<br /><br />Therefore, the correct answer is $\frac{0.5^{3x+5}}{3 \ln(0.5)} + C$.