This book is specifically designed for students preparing for competitive exams such as NET (JRF), GATE, and other advanced mathematics entrance tests. It provides a rigorous and detailed exploration of core topics in integral calculus, with a focus on concepts that frequently appear in such exams. The chapters are structured to build a deep theoretical understanding while offering practical techniques that can be applied to problem-solving in these competitive settings.

In Chapter 1, we begin with Riemann sums and the Riemann integral, two essential concepts in calculus. The chapter covers partitioning of intervals and upper and lower Riemann sums, culminating in the Fundamental Theorem of Calculus, which forms a critical link between differentiation and integration—topics often tested in competitive exams.Chapter 2 delves into improper integrals, a topic of significant importance in many entrance exams. We focus on convergence and divergence of integrals over infinite intervals or with unbounded integrands, along with techniques like the Comparison Test and Limit Comparison Test, ensuring that students gain the tools needed to tackle challenging integrals.Chapter 3 focuses on monotonic functions, covering their continuity, differentiability, and behavior, all of which are key in understanding more advanced mathematical functions.In Chapter 4, we address types of discontinuities—a crucial topic for competitive exams. We explore the classification of discontinuities and examine the behavior of functions near these points.

This book provides a comprehensive, exam-focused approach to mastering integral calculus, preparing students for high-level competitive exams in mathematics.