This introductory chapter lays the foundation for the course by defining the concept of integration and its relationship to differentiation. Students will learn the basic properties of integrals, the notation and terminology used in integral calculus, and the significance of the fundamental theorem of calculus.

This chapter delves into the various techniques used to evaluate indefinite integrals, including the power rule, the method of substitution, integration by parts, and integration of trigonometric functions. Students will explore the step-by-step processes for applying these techniques and will develop the ability to select the appropriate method based on the given function or expression.

In this chapter, students will learn about definite integrals, their properties, and the various methods for evaluating them. They will explore the concept of the Riemann sum and its relationship to the definite integral, as well as the application of the fundamental theorem of calculus to simplify the calculation of definite integrals. Additionally, students will learn about the properties of definite integrals, such as linearity, the change of variable formula, and integration by parts, equipping them with the tools to solve a diverse range of integration problems.

This chapter focuses on the practical applications of definite integrals, highlighting their importance in solving real-world problems. Students will learn how to use definite integrals to calculate areas, volumes, and other geometric properties. They will also explore the use of integrals in determining the work done by a variable force, the arc length of a curve, and the center of mass of a planar region. By engaging with these applications, students will develop a deeper appreciation for the versatility and power of integral calculus in various fields, including physics, engineering, and economics.

In this chapter, students will delve into the concept of improper integrals, where the limits of integration may be infinite or the integrand may be undefined at certain points. They will learn the techniques for evaluating these integrals, as well as the criteria for determining their convergence or divergence. This understanding will enable students to handle a broader range of integration problems, including those involving singularities, asymptotes, and infinite domains.

The final chapter of the course explores the relationship between integration and differential equations, highlighting the role of integrals in solving various types of differential equations. Students will learn how to use integration techniques to find the general and particular solutions of first-order differential equations, as well as how to apply integration to model and analyze real-world phenomena described by differential equations. This chapter will equip students with the skills to tackle problems in areas such as physics, engineering, and the life sciences, where the integration of functions is essential for understanding and predicting dynamic systems.