This beginners' course provides students with a general and sufficiently easy to grasp theory of the Kurzweil-Henstock integral. The integral is indeed more general than Lebesgue's in R^N, but its construction is rather simple, since it makes use of Riemann sumswhich, being geometrically viewable, are more easy to be understood.The theory is developed also for functions of several variables, and for differential forms, as well, finally leading to the celebrated StokesCartan formula. In the appendices, differential calculus in R^Nis reviewed, with the theory of differentiable manifolds. Also, the BanachTarski paradox is presented here, with a complete proof, a rather peculiar argument for this type of monographs.