Exercise 14

Consider the indefinite integral given by $\int e^{2x} \cos(e^x) \, dx$.

1. Noting that $e^{2x} = e^x \cdot e^x$, use the substitution $z = e^{x}$ to determine a new, equivalent integral in the variable $z$.

2. Evaluate the integral you found in (a) using an appropriate technique.

3. How is the problem of evaluating $\int e^{2x} \cos(e^{2x}) \, dx$ different from evaluating the integral in (a)? Do so.

4. Evaluate each of the following integrals as well, keeping in mind the approach(es) used earlier in this problem:

   • $\int e^{2x} \sin(e^x) \, dx$

   • $\int e^{3x} \sin(e^{3x}) \, dx$

   • $\int xe^{x^2} \cos(e^{x^2}) \sin(e^{x^2}) \, dx$