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.