Consider the indefinite integral given by ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{2x}cos(e^x)dx.$

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

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

c$.$ How is the problem of evaluating ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{2x}cos(e^{2x})dx$ different from evaluating the integral in $($a$)?$ Do so$.$

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

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{2x}sin(e^x)dx$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{3x}sin(e^{3x})dx$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x{e}^{x^2}cos(e^{x^2})sin(e^{x^2})dx$