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Thread: Please help with a problem on fundamental theorems of calculus?

  1. #1
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    Please help with a problem on fundamental theorems of calculus?

    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!
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  2. #2
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    Re: Please help with a problem on fundamental theorems of calculus?

    Quote Originally Posted by kaonashi View Post
    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, $\displaystyle\int_a^b {{e^{x + c}}dx} = {e^c}\int_a^b {{e^x}dx} = {e^c}\left( {{e^b} - {e^a}} \right) = \left( {{e^{b + c}} - {e^{a + c}}} \right)$
    Thanks from kaonashi and topsquark
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