Use the hint to express the integral as shown below.
[tex]\begin{gathered} \int_{1}^{3} \frac{2^{\log_3 (x^3 + 2x^2)}}{x \cdot 4^{\log_3 \sqrt{x+2}}} \, dx = \int_{1}^{3} \frac{(x^3 + 2x^2)^{\log_3 2}}{x \cdot (\sqrt{x+2})^{\log_3 4}} \, dx \end{gathered}[/tex]
Remember that log₃(4) = 2log₃(2) and use the power of a power law of exponent to turn the latter integral as below.
[tex]\begin{gathered}\int_{1}^{3} \frac{(x^3 + 2x^2)^{\log_3 2}}{x \cdot (\sqrt{x+2})^{\log_3 4}} \, dx = \int_{1}^{3} \frac{1}{x}\left(\frac{x^3 + 2x^2}{(\sqrt{x+2})^{2}} \right)^{\log_3 2} \, dx \end{gathered}[/tex]
You can do the rest. I am sure you know how to simplify the fraction inside the parentheses. You should be able to get:
[tex]\begin{gathered} \int_{1}^{3} \frac{x^{2\log _3 2}}{x} \, dx = \int_{1}^{3} x^{\log _3 4-1}\, dx = \left.\frac{x^{\log _3 4}}{\log _3 4}\right|_{1}^{3}\end{gathered}[/tex]
This simplifies to 3/log₃(4) = log(27)/log(4).
