_Integration of Transcendental Functions._ The truly analytical integration of transcendental functions is as yet very little advanced, whether for _exponential_, or for _logarithmic_, or for _circular_ functions. But a very small number of cases of these three different kinds have as yet been treated, and those chosen from among the simplest; and still the necessary calculations are in most cases extremely laborious. A circumstance which we ought particularly to remark in its philosophical connection is, that the different procedures of quadrature have no relation to any general view of integration, and consist of simple artifices very incoherent with each other, and very numerous, because of the very limited extent of each. One of these artifices should, however, here be noticed, which, without being really a method of integration, is nevertheless remarkable for its generality; it is the procedure invented by John Bernouilli, and known under the name of _integration by parts_, by means of which every integral may be reduced to another which is sometimes found to be more easy to be obtained. This ingenious relation deserves to be noticed for another reason, as having suggested the first idea of that transformation of integrals yet unknown, which has lately received a greater extension, and of which M. Fourier especially has made so new and important a use in the analytical questions produced by the theory of heat. _Integration of Algebraic Functions._ As to the integration of algebraic functions, it is farther advanced. However, we know scarcely any thing in relation to irrational functions, the integrals of which have been obtained only in extremely limited cases, and particularly by rendering them rational. The integration of rational functions is thus far the only theory of the integral calculus which has admitted of being treated in a truly complete manner; in a logical point of view, it forms, then, its most satisfactory part, but perhaps also the least important. It is even essential to remark, in order to have a just idea of the extreme imperfection of the integral calculus, that this case, limited as it is, is not entirely resolved except for what properly concerns integration viewed in an abstract manner; for, in the execution, the theory finds its progress most frequently quite stopped, independently of the complication of the calculations, by the imperfection of ordinary analysis, seeing that it makes the integration finally depend upon the algebraic resolution of equations, which greatly limits its use.