### 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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