an integral is a certain limit. you can read up what this limit is in any calculus book and you should. doing, or solving or whatever, to the integral involves finding or evaluating that limit.
there are often no practical ways to find this limit, as is usual with real life limits. But in many interesting cases these limits can be found in various ways. Archimedes managed to find some, enough to compute the volume of a ball.
The calculus book by Apostol is excellent for giving several interesting and non trivial examples of integrals which can be computed by hand.
A few hundred years ago it was noticed that if the integral is looked at as defining a function, then there is a certain differential equation satisfied by this function. this is called the fundamental theorem of calculus
That implies that if we can solve the differential equation, we can also evaluate the integral. In many many specific cases these solutions, called "antiderivatives" have been written down in tables and can be looked up.
But it is important to understand that in general there is no known explicit solution to the differential equation, and in fact the only way to define a solution, is to use the limit definition of the integral. e.g. even the integral of a simple function like cos(x^2) apparently cannot be evaluated by the FTC.
So books that discuss differentiation first and then use only the fundamental theorem to evaluate integrals do a disservice, since those students never appreciate what an integral is, nor the fact that most of the time in the real world, the theorem is of no use.
Again, if you want to understand integrals, and calculus in general, read a book like apostol.
