You'll want to understand the different techniques for different types of integrals. For instance, simple polynomials can be integrated very easily, whereas a product of functions has a special technique called "Integration by Parts" that is used to solve the integral. It simply depends on the format of the integrand (what is inside the integral).
No
No, all functions are not Riemann integrable
If you mean the the integral of sin(x2)dx, It can only be represented as an infinite series or a unique set of calculus functions known as the Fresnel Integrals (Pronounced Frenel). These functions, S(x) and C(x) are the integrals of sin(x2) ans cos(x2) respectively. These two integrals have some interesting properties. To find out more, go to: http://en.wikipedia.org/wiki/Fresnel_integral I hope this answers your question.
The first thing that come up into my mind is numbers, calculation, integrals and derivatives
You could look at the length of the walk and use integrals to determine that.
Flux integrals, surface integrals, and line integrals!
A. M. Bruckner has written: 'Differentiation of integrals' -- subject(s): Integrals
No
Yes, but only in some cases and they are special types of integrals: Lebesgue integrals.
Gottfried Wilhelm Leibniz is credited with defining the standard notation for integrals.
There are two types of integrals: definite and indefinite. Indefinite integrals describe a family of functions that differ by the addition of a constant. Definite integrals do away with the constant and evaluate the function from a lower bound to an upper bound.
Stanislaw Hartman has written: 'The theory of Lebesgue measure and integration' -- subject(s): Generalized Integrals, Integrals, Generalized
Richard M. Hain has written: 'Iterated integrals and homotopy periods' -- subject(s): Homotopy theory, Multiple integrals
D. C. Khandekar has written: 'Path-integral methods and their applications' -- subject(s): Path integrals, Feynman integrals
There are many websites that contain information on how Integrals work in calculus. Among them are Tutorial Math, Wolfram, Ask A Mathematician, and Hyper Physics.
Jean Marie Defenbach has written: 'A review of several classical measure functions' -- subject(s): Generalized Integrals, Integrals, Generalized
Ralph Calvin Applebee has written: 'A two parameter Laplace's method for double integrals' -- subject(s): Integrals, Laplace transformation