The integral of x^x can not be expressed using elementary functions. In fact, this is true about many integrals.
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.
Flux integrals, surface integrals, and line 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.
William Henry Maltbie has written: 'On the curve ym-G(x)=0, and tis associated Abelian integrals ..' -- subject(s): Accessible book 'On the curve y[superscript m] - G(x) = 0, and its associated Abelian Integrals'
A. M. Bruckner has written: 'Differentiation of integrals' -- subject(s): Integrals
Gottfried Wilhelm Leibniz is credited with defining the standard notation for integrals.
Yes, but only in some cases and they are special types of integrals: Lebesgue integrals.
polynomial functions: x2,x3,x1/2,x-1,xn if n doesn't equal 1 or 0 any trigonometric function: sin(x), cos(x), tan(x), sec(x), csc(x), cot(x) exponential functions: 2x,3x,.5x,ex,nx if n doesn't equal 1 or 0. any logarithmic function: log10(x), log2(x), ln(x), logn(x), for every n. any inverse trig function: sin-1(x),sin-1(x),cos-1(x),tan-1(x),sec-1(x),csc-1(x),cot-1(x) hyperbolic functions and their inverses: sinh(x),cosh(x),tanh(x),sech(x),csch(x),coth(x), sinh-1(x),cosh-1(x),tanh-1(x),sech-1(x),csch-1(x),coth-1(x) Hypergeometric function, Logarithmic integrals, Error functions, Fresnel integrals, Elliptic Integrals, and the integrals of nearly every other function. Also, any combination of the above functions.
A line integral is a simple integral. they look like: integral x=a to b of (f(x)). A surface integral is an integral of two variables. they look like: integral x=a to b, y=c to d of (f(x,y)). or integral x=a to b of (integral y=c to d of (f(x,y))). The second form is the nested form. A pair of line integrals, one inside the other. This is the easiest way to understand surface integrals, and, normally, solve surface integrals. A volume integral is an integral of three variables. they look like: integral x=a to b, y=c to d, z=e to f of (f(x,y,z)). or integral x=a to b of (integral y=c to d of (integral z=e to f of f(x,y,z))). the above statement is wrong, the person who wrote this stated the first 2 types of integrals as regular, simple, scalar integrals, when line and surface integrals are actually a form of vector calculus. in the previous answer, it is stated that the integrand is just some funtion of x when it is actually usually a vector field and instead of evaluating the integral from some x a to b, you will actually be evaluating the integral along a curve that you will parametrize to get the upper and lower bounds of the integral. as you can see, these are a lot more complicated. looking at your question tho, i dont think you want the whole expanation on how to solve these problems, but more so what they are and what they are used for, because these can be a pain to solve and there are also several ways to solve them indirectly. line integrals have an important part in physics because they alow us to calculate things such as work that have vector values rather than just scalar values as you can use these integrals to describe a particles path along a curve in a force field. surface integrals help us calculate things like flux, or how fluid flows over a surface. if you want to learn more, look into things like greens theorem, or the divergence theorem. p.s. his definition of a surface integral is acutally how you find the volume of a region