I didn't immediately answer this because if it is about actually solving general integral equations it is beyond what I have experience with. But some of the responses referred to the straight evaluation of integrals and that I can say something about.
The Presentations package has a subsection called Student's Integral that allows the step by step evaluation of definite or indefinite integrals. This contains commands for operating on an integral expression. It doesn't just show the steps that Mathematica might (or might not) use but allows the student (or researcher) to specify the steps to be used. The sub-package comes with a BasicIntegralTable, such as students might use, or allows users to construct their own integral table. Alternatively at any point an integral can be turned over to the regular Mathematica Integrate or NIntegrate command.
The user writes an integrate command (with a small i) that keeps it in an unevaluated form, and then operates on it by various commands. The user can operate on the integrand, say by using Apart or CompletTheSquare. Integral sums can be broken our and constants taken outside of the integral. The user can use change of variable, integration by parts or trigonometric substitution. Integration by parts throws off a panel of information showing the internal substitutions. Trigonometric substitution throws off a triangle diagram showing the expressions on each side. Definite integrals return a LimitsBracket expression such as is usually seen in textbooks that can then be evaluated.
Peter Lindsay at the Mathematical Institute in the University of St Andrews [ www.mcs.st-and.ac.uk ] has kindly undertaken to maintain an archive that provides downloadable notebooks and PDF files for various Presentations solutions that have appeared on MathGroup.
A Step by Step Integration example for the volume of a surface of revolution using integration by parts will be shown there.
Mathematica does many things automatically and most users prefer it that way. For teaching purposes this is often a problem. Nevertheless, it is possible to work around it and create definitions and possibly formatting that allows students to work conveniently at a more basic level. The developers of Mathematica can't get every convenient thing, for every field, in there. But that doesn't mean that Mathematica doesn't have the capability to tackle these kinds of applications. It is just that users and third party developers have to develop some of these applications.
Mathematica is not so much a tool for doing mathematics as a meta-tool for making the tools to conveniently do mathematics on specific applications.
I have such a question: Is it possible to display in Mathematica the solution for example of integral equation step by step? Not only the final solution. If yes, how to do it? I've not found anything about this in Documentation Center unless I missed it.