1. The problem is to find the integral $$\int a^{3^{\log_a x}} \, dx$$ plus the constant of integration $c$. 2. First, simplify the exponent: note that $$3^{\log_a x}$$ is a bit unusual, but we can rewrite it using properties of logarithms and exponents. 3. Recall that $$a^{\log_a x} = x$$ by definition of logarithm. 4. However, here the exponent is $$3^{\log_a x}$$, which is not the same as $$a^{\log_a x}$$. 5. Let's rewrite $$3^{\log_a x}$$ using the change of base formula: $$3^{\log_a x} = e^{\ln(3) \cdot \log_a x} = e^{\ln(3) \cdot \frac{\ln x}{\ln a}} = x^{\frac{\ln 3}{\ln a}}$$ 6. Therefore, the integrand becomes: $$a^{3^{\log_a x}} = a^{x^{\frac{\ln 3}{\ln a}}}$$ 7. This expression is complicated, but the problem likely intends a simpler interpretation. Possibly the problem means: $$\int a^{3 \log_a x} \, dx$$ 8. Using the property of logarithms: $$a^{3 \log_a x} = (a^{\log_a x})^3 = x^3$$ 9. So the integral simplifies to: $$\int x^3 \, dx$$ 10. Integrate: $$\int x^3 \, dx = \frac{x^4}{4} + c$$ 11. Thus, the answer is option (d) $$\frac{1}{4} x^4$$ plus the constant $c$. Final answer: $$\int a^{3 \log_a x} \, dx = \frac{1}{4} x^4 + c$$