This textbook entitled "An introduction to Calculus of variations and Integral equations" is intended to study the extremals of different types of variational problems and methods of finding the explicit solutions of integral equations, where ever possible. The absence of methods of finding an exact solution is intended to study the properties of solutions of the given integral equations. This book contains a total of 07 chapters and two sections. section-I includes the calculus of variation, while section-II discusses the part of the Integral Equation. Section-I has been divided into four chapters, while section-II has been divided into 03 chapters. This book is based on the syllabi of the theory of Calculus of variations and Integral equations prescribed for postgraduate students of mathematics and applied mathematics in different institutions like N.I.T¿s, I.I.T¿s, and universities of India abroad. This book will be useful for competitive examinations as well.