So that is not your work but someone else's that you are asking about?
You should know that the derivative of \(\displaystyle e^x\) is just \(\displaystyle e^x\) again so the anti-derivative of \(\displaystyle e^x\) is \(\displaystyle e^x+ C\). The derivative of \(\displaystyle sin(x)\) is \(\displaystyle cos(x)\) so the anti-derivative of \(\displaystyle cos(x)\) is \(\displaystyle sin(x)+ C'\) and the derivative of \(\displaystyle cos(x)\) is \(\displaystyle -sin(x)\) so the anti-derivative of \(\displaystyle cos(x)\) is \(\displaystyle -sin(x)+ C''\). (C, C', and C'' are constants, not necessarily the same.)
