What is the physical significance of integration and why can't certain functions don't have indefinite integral but definite integral exist?

When we approached integral calculas in high school first indefinite integration and then definite Integration were taught. And some integration doesn't have any significance in indefinite integration but in definite it exist. Like integration ofe^-x^2 does not exist indefinitely but do exist as definite integral So why is that? what is it's physical significance?

There is no physical significance to what we think of as elementary functions. $e^{-x^2}$ has an antiderivative, it is just not an elementary function. Other non-elementary functions appear in applications too; the first that come to mind are the elliptic integrals and the Bessel functions. As for functions with no elementary antiderivative which nevertheless have a closed form for certain definite integrals, this is actually a poorly understood topic. There is no known counterpart to the Risch algorithm (a procedure that can be used to find elementary antiderivatives or prove that none exist) for definite integration.

• Thank you so much for the beautiful explanation. But can you give me some source from where I can get theories regarding indefinite integration to benefit my concept? – Demietra95 May 31 '15 at 13:19
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