Answer
$ \dfrac{-1}{4} \cos (4 x) +C $
Work Step by Step
Our aim is to solve the integral $ \int e^{2x} \cos (e^{2x}+1) \ dx$
Use formula:$ \int \sin (ax+b) \ dx =\dfrac{-1}{a} \cos (ax+b)+C$
Let us consider that $a =4$ and $b=0$
Now, $ \int e^{2x} \cos (e^{2x}+1) \ dx=\dfrac{-1}{4} \cos (4 x) +C $
or, $= \dfrac{-1}{4} \cos (4 x) +C $
