Answer
$ \dfrac{1}{2 } \sin (e^{2x}+1) +C$
Work Step by Step
Our aim is to solve the integral $ \int e^{2x} \cos (e^{2x}+1) \ dx$
Let us consider that $a =(e^{2x}+1)$ and $\dfrac{da}{dx}=2 e^{2x} \implies dx=\dfrac{1}{2 e^{2x}} \ da$
Now, $ \int e^{2x} \cos (e^{2x}+1) \ dx = \int e^{2x} \cos a (\dfrac{1}{2 e^{2x}} ) \ dx$
or, $= \dfrac{1}{2 } \int \cos a \ da$
or, $= \dfrac{1}{2 } \sin a +C$
or, $= \dfrac{1}{2 } \sin (e^{2x}+1) +C$
