Well in the first method, you have left off 1/2 until after the "v" substitution. But I expect that might just have been cut off...

But then you left off the negative...

But then that's back later...

Go back and fix up all your sloppy errors.

In the second, what your classmate did is valid. Surely you can see that $\displaystyle \begin{align*} \mathrm{e}^{2x} + 4 - \mathrm{e}^{2x} = 4 \end{align*}$, so 1/4 of that is 1, as required.