Hi everyone. My friend has showed me how to solve this particular problem (attached picture) and the solution (also attached) is, apparently, pretty simple. Despite that, I don't understand why the indefinite integral of ex is still ex. Aren't you supposed to add 1 to the exponent and then put that same value on the denominator? I don't get why its integral is the same... Please clear my confusion.

Thank you very much for your help!

Originally Posted by kaonashi
Hi everyone. My friend has showed me how to solve this particular problem (attached picture) and the solution (also attached) is, apparently, pretty simple. Despite that, I don't understand why the indefinite integral of ex is still ex. Aren't you supposed to add 1 to the exponent and then put that same value on the denominator? I don't get why its integral is the same... Please clear my confusion.
Surely you know the the derivative of $e^x$ is $e^x~?$
Therefore $\int {{e^x}dx} = {e^x}$

Moreover, $\displaystyle\int_a^b {{e^{x + c}}dx} = {e^c}\int_a^b {{e^x}dx} = {e^c}\left( {{e^b} - {e^a}} \right) = \left( {{e^{b + c}} - {e^{a + c}}} \right)$