Nonelementary Integral Related Question

[megatyler30]

Recently, I have looked into nonelementary integrals and I have a question.

When looking at ∫sin(sinx)dx, which I know cannot be represented as an elementary function, I wondered what the function would look like. Using mathematica, I was able to get a graph of f(x)=∫₀^xsin(sinx)dx. Does the represent the same function as ∫sin(sinx)dx? And more importantly, why? (Sorry about the bad formating for limits of integration)

Edit: From what I've seen, it is if and only if the function g(x)=∫sin(sinx)dx is 0 at x=0. If this was true, then how would one go about proving if g(0)=0 or not?

 

[mathman]

You have one integral with limits 0 and x and another integral without limits. You need to clarify the relationship.

 

[megatyler30]

I am asking about the relationship between the two. The edit was from what I've seen from posts on here but I don't know if that's the relationship (may be incorrect or I may have interpreted the posts incorrectly).

Edit: Looked one of the posts over again.
In this case: f(x)=g(x)-g(0)
So still it depends on if g(0) is 0 or not.
If g(0)=0 then f(x)=g(x) if not then f(x)=g(x)+c where c=-g(x)
Is this thought process correct?

 

[mfb]

I am asking about the relationship between the two.

Yes, but the question does not make sense. It is like asking about the relationship between x and 3.
An integral without limits does not have a value, an integral with limits does.

 

[megatyler30]

Okay let me clarify. I'm defining a function f(x) such that f(x)=∫0xsin(sinx)dx. This is no different than saying f(x)=∫0xsin(sin(t))dt since the variable in the function that the integral is being taken of is basically just a dummy variable, so t would be more clear. For example ∫0xsin(sint)dt=g(x)-g(0). So as an example, f(1)=∫0 ¹sin(sin(t))dt=g(1)-g(0) which IS a value. Anyways I guess by having to explain it both here and the edit, I figured out the answer to my question.

See the error function to see an example of a similarly defined function.

 

[HallsofIvy]

\int sin(sin(x))dx is NOT a single function. It is a "family" of functions, all differing by numbers, not functions of x. \int_0^x sin(sin(x))dx is one of those functions.

 

[megatyler30]

Yeah, I figured it out when explaining it to the other posters.

I did want some confirmation though, so thanks for the confirmation!