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How do I show that a function defined by an integral is of class C1?

  1. Mar 26, 2013 #1
    1. The problem statement, all variables and given/known data

    [tex]F(x)=\int^{2x}_1\frac{e^{xy}\cos y}{y}dy[/tex]

    Show that F is of class [itex]C^1[/itex], and compute the derivative F′(x).

    2. Relevant equations

    Thm:

    Suppose S and T are compact subsets of [itex]\mathbb{R}^n \text{ and } \mathbb{R}^m[/itex], respectively, and S is measurable. if [itex]f(\bf{x,y})[/itex] is continuous on the set [itex]T\times S = \{ (\bf{x,y})\; : \; \bf{x}\in T, \;\bf{y}\in S\}[/itex], then the function F defined by, [tex]F(x)=\int ... \int_S f(x,y)d^n\bf{y}[/tex] is continuous on T.

    Thm:

    Suppose [itex] S\subset \mathbb{R}^n \text{ and } \bf{f}\; : \; S\rightarrow\mathbb{R}^m[/itex] is continuous at every point of S. If S is compact, then [itex]\bf{f}[/itex] is uniformly continuous on S.

    3. The attempt at a solution

    I already figured out that

    [tex]\int e^{xy}\cos y\;dy=\frac{e^{xy}}{x^2+1}(\sin y+x\cos y)[/tex]

    which was listed as a hint. My book explains how to compute the derivative, but not how to show that it is of class [itex]C^1[/itex]. I need some help getting started on this one!
     
  2. jcsd
  3. Mar 26, 2013 #2
    When you differentiate a [itex] C^k [/itex] function you get a [itex] C^{k-1} [/itex] function. What happens when you integrate a [itex] C^k [/itex] function?
     
  4. Mar 26, 2013 #3
    uh, i'm gonna go with [itex]C^{k+1}[/itex]
     
  5. Mar 26, 2013 #4
    are you sure about that though?
     
  6. Mar 26, 2013 #5
    If [itex] f \in C^k(\mathbb R, \mathbb R) [/itex] then define [itex] g(x) = \int_1^x f(t) gt [/itex]. We would like to show that [itex] g \in C^{k+1} [/itex], and hence that the (k+1) derivatives of g exist and are continuous. Well, [itex] \frac{d^{k+1} g}{dx^{k+1}} = \frac{d^k f}{dx^k} [/itex] and this is continuous by assumption that [itex] f \in C^k [/itex].
     
  7. Mar 26, 2013 #6
    For multivariate calculus, do the same argument with partials. Though here you need not do that since your function is just a map [itex] F: \mathbb R \to \mathbb R [/itex].

    Now the solution is not quite straightforward: you are going to need to check that the derivative of your F(x) is continuous. You can do this using the Leibniz rule.
     
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