This book has been written as a companion volume to my Treatise on the Differential Calculus, and in its construction I have endeavoured to carry out the same general plan on which that book was composed. I have, accordingly, studied simplicity so far as was consistent with rigour of demonstra tion, and have tried to make the subject as attractive to the beginner as the nature of the Calculus would permit. I have, as far as possible, confined my attention to the general principles of Integration, and have endeavoured to arrange the successive portions of the subject in the order best suited for the Student.