Chapter 7: Integrals

7.1 Overview

7.1.1 Indefinite Integrals

• If ( \frac{d}{dx} F(x) = f(x) ), then ( \int f(x) dx = F(x) + C ).
• These are called indefinite integrals, where ( C ) is the constant of integration.
• Indefinite integrals differ by a constant.

7.1.2 Function Derivatives

• Two functions differing by a constant have the same derivative.

7.1.3 Geometric Interpretation

• ( \int f(x) dx = F(x) + C = y ) represents a family of curves.
• Different values of ( C ) correspond to different curves in the family.

7.1.4 Properties of Indefinite Integrals

• Differentiation and integration are inverses.
  • ( \frac{d}{dx}\int f(x)dx = f(x) )
  • ( \int f'(x)dx = f(x) + C )
• Indefinite integrals with the same derivative are equivalent.
• Integral of the sum of functions equals the sum of their integrals.
• A constant factor can be rearranged with the integral sign.
• Properties generalized for finite functions and real numbers.

7.1.5 Methods of Integration

• Integration by substitution.
• Integration using partial fractions.
• Integration by parts.

7.1.6 Definite Integrals

• Denoted by ( \int_a^b f(x)dx ).
• Evaluated as:
  • The limit of a sum.
  • ( \int_a^b f(x)dx = F(b) - F(a) ) if ( F ) is an antiderivative.

7.1.7 Definite Integral as a Limit of a Sum

• ( \int_a^b f(x)dx ) is the area bounded by the curve ( y = f(x) ).

7.1.8 Fundamental Theorem of Calculus

• First Theorem: If ( f ) is continuous on ( [a, b] ) and ( A(x) ) is the area function, then ( A'(x) = f(x) ).
• Second Theorem: If ( f ) is continuous and ( F ) an antiderivative, then ( \int_a^b f(x)dx = F(b) - F(a) ).

7.1.9 Properties of Definite Integrals

• ( \int_a^a f(x)dx = 0 ).
• ( \int_a^b f(x)dx = \int_b^a f(x) ).
• Continuous sum property: ( \int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx ).
• Symmetry properties for even and odd functions.

7.2 Solved Examples

• Various examples provided to illustrate integration techniques.
• Examples demonstrate substitution, parts, and evaluation as limits of sums.
• Complex functions are broken down into simpler forms for integration.

7.3 Exercises

• Short Answer (S.A.) problems to verify integration properties.
• Evaluation of integrals using different techniques.
• Long Answer (L.A.) problems requiring detailed solutions.

Objective Type Questions

• Exercises with multiple-choice questions.
• Completion of mathematical expressions based on integration principles.
• Fill in the blanks and true/false type questions to test comprehension.

---

Note:

• All examples and exercises focus on applying the theoretical concepts of integrals.
• Emphasis on understanding the properties and applications of both indefinite and definite integrals.
• The chapter is structured to build a foundational understanding of integration in calculus.