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

[richyw]

Homework Statement

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

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

Homework Equations

Thm:

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

Thm:

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

The Attempt at a Solution

I already figured out that

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

which was listed as a hint. My book explains how to compute the derivative, but not how to show that it is of class C^1. I need some help getting started on this one!

 

[Kreizhn]

When you differentiate a C^k function you get a C^{k-1} function. What happens when you integrate a C^k function?

 

[richyw]

uh, I'm going to go with C^{k+1}

 

[richyw]

are you sure about that though?

 

[Kreizhn]

If f \in C^k(\mathbb R, \mathbb R) then define g(x) = \int_1^x f(t) gt. We would like to show that g \in C^{k+1}, and hence that the (k+1) derivatives of g exist and are continuous. Well, \frac{d^{k+1} g}{dx^{k+1}} = \frac{d^k f}{dx^k} and this is continuous by assumption that f \in C^k.

 

[Kreizhn]

For multivariate calculus, do the same argument with partials. Though here you need not do that since your function is just a map F: \mathbb R \to \mathbb R.

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.