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It is well known that the operation of integration may lead to divergent formal expansions like as soon as one leaves the area of formal power series for the area of formal transseries. On the other hand, from the analytic point of view, the operation of integration is usually “regularizing”, in the sense that it improves convergence rather than destroying it. For this reason, it is natural to consider so called “integral transseries” which are similar to usual transseries except that we are allowed to recursively keep integrals in the expansions. Integral transseries come with a natural notion of “combinatorial convergence”, which is preserved under the usual operations on transseries, as well as integration. In this paper, we lay the formal foundations for this calculus.