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Homework Help: How do you differentiate & integrate e to the x?

  1. Sep 6, 2009 #1
    1.a Evaluate the following. Show all work.
    http://img90.imageshack.us/img90/289/problemu.png [Broken]
    1.b Differentiate.
    http://img293.imageshack.us/img293/1440/problem2.png [Broken]

    2. Relevant equations
    ƒ b-a f(x)-g(x)

    3. The attempt at a solution

    1.a.http://img511.imageshack.us/img511/1205/problemews.png [Broken]

    1.b.http://img512.imageshack.us/img512/4250/problem2ans.png [Broken]
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Sep 6, 2009 #2
    e^2x does not integrate to 2e^2x because differentiation of 2e^2x is 4e^2x which is not equal to the e^2x.
  4. Sep 6, 2009 #3


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    Presumably, you know that [itex](e^x)'= e^x[/itex] and [itex]\int e^x dx= e^x+ C[/itex].

    To differentiate [itex]e^{2x}[/itex], use the chain rule, df/dx= (df/du)(du/dx), with u= 2x. Then [itex]f(u)= e^u[/itex] so [itex]df/du= e^u[/itex] and u= 2x so du/dx= 2.

    To integrate [itex]e^{2x}[/itex], use the "inverse" of the chain rule- substitution. Let u= 2x so du= 2dx or dx= (1/2)du.

    The one multiplies by 2, the other divides by 2.
  5. Sep 6, 2009 #4
    So [itex]e^2x[/itex] differentiated would become [itex]2e^2x[/itex]?
  6. Sep 6, 2009 #5


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    Homework Helper

    (I assume you mean e2x differentiated would become 2e2x?)

    Yes, that's right. :smile:

    (To see why, apply the chain rule with f(x) = ex and g(x) = 2x.)
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