Let's go through each problem one by one.

Problem 1

The integral to solve is:

Looking closely, this integral seems to be designed to test simplification techniques rather than actual integration since it contains logarithmic terms and exponents in a complex way. However, by applying properties of logarithms, let's examine each option one by one to check if there’s a matching form after simplification.

After carefully analyzing the choices, option (b) is likely the correct answer:

Problem 2

The second question asks for the limit:

This expression can be analyzed using series and limits. As $n \to \infty$, each term in the series approximates to $\frac{1}{\sqrt{2n^2}}$ or similar simplifications. By evaluating this series, the limit simplifies to option (a):

Final Answers:

1. Answer: (b) $a^x \ln \left( \frac{x}{e} \right)^x + C$
2. Answer: (a) $\sqrt{2} - 1$

Would you like a detailed solution for each part, or do you have further questions?

Here are some questions that might expand your understanding:

1. How do properties of logarithms and exponents help simplify complex integrals?
2. What techniques can simplify series expressions when approaching infinity?
3. What are some common mistakes to avoid in limit problems involving infinity?
4. How can we identify when an integral is a standard form versus when it requires unique transformations?
5. In what situations can logarithmic and exponential functions interchange roles in calculus?

Tip: Always check if the integral or limit can be simplified using basic logarithmic or exponential properties before attempting complex techniques.