Integral Evaluation. How can we justify the fact that some integrals can't be evaluated? It's like we can't sum up a function within two bounds or we are unable to find the area under the curve of a function. How's that possible?
 A: Actually, Integral is infinite series. It is not area, but you if you approximate area by using respectively small rectangles of random number (or maybe some trapezoid or parabolas) then you will get some result. That is with some error, approximation to area under the curve. On the other hand if you sum up length of line segments of random number, you will get approximation for length of curve. Now lets go back to area example. As you increase the number of rectangles, your approximation will be more precise. Then you may want to try infinitely many rectangles for approximation. That is what we do for computing areas. Still you need to sum these areas one by one by hand. 
But we are lucky. Fundamental theorem of calculus says that if you can find antiderivative of a function then you do not need to sum these areas one by one. You can use antiderivatives to evaluate that sum.
If there is no antiderivative in terms of known elementary functions, you will need to sum these series by hand, which is what you think that can't be evaluated. There is still many other ways to find areas by different approximations. Google about trapezoid, simpson rules for these approximations. 
Hope it helps.
