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Integration of e^x = ...?

____________________________________

Answer
      x   
     e
    ---- + C
    ln e
Explanation

We know the integration formula of "a raise to power x".

    /             x
    |  x        a
    | a  dx = ------  + C  for base 10 logarithm     .....(i)
    |          log a  
    /

Similarly for "e raise to power x". 

    /            x
    |  x        e
    | e  dx = ------ + C  for base e logarithm     ..... (ii)
    |          ln e
    /

We should memorize the formulas (i) & (ii) because it is easy to remember, clear the basic concept and theam of boths result are also same.

Here because ln e = 1 (because e^0 = 1 and by use logarithm property we get ln e = 1 ) after simplification we can get

         x        x
        e        e
    = ------ = ------ = e^x
       ln e      1

    /  x       x           
    | e  dx = e  + C                          ......... (iii)   
    /

Integration formulas (ii) & (iii) are same formula. We mostly use formula (iii) because it is understood that "ln e = 1". Just keep in mind the original formula (ii) and there is also the hidden knownledge that "ln e = 1".

____________________________________

    /            x
    |  x        e
    | e  dx = ------ + C       (here  ln e = 1)
    |          ln e
    /
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