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Introduction to Integral Calculus by Rohde and GC Jain

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Introduction to Integral Calculus (Systematic Studies with Engineering Applications for Beginners) written by Ulrich L. Rohde, Prof. Dr.-Ing. Dr. h. c. mult., BTU Cottbus, Germany, Synergy Microwave Corporation Peterson, NJ, USA G. C. Jain (Retd. Scientist) Defense Research and Development Organization, Maharashtra, India Ajay K. Poddar, Chief Scientist, Synergy Microwave Corporation, Peterson, NJ, USA A. K. Ghosh, Professor, Department of Aerospace Engineering, Indian Institute of Technology – Kanpur, Kanpur, India The purpose of these works is to provide the basic (but solid) foundation of Calculus to beginners. The books aim to show them the enjoyment in the beauty and power of Calculus and develop the ability to select proper material needed for their studies in any technical and scientific field, involving Calculus
The author’s aim throughout has been to provide a tour of Calculus for a beginner as well as strong fundamental basics to undergraduate students on the basis of the following questions, which frequently came to our minds, and for which we wanted satisfactory and correct answers.
(i) What is Calculus?
(ii) What does it calculate?
(iii) Why do teachers of physics and mathematics frequently advise us to learn Calculus seriously?
(iv) How is Calculus more important and more useful than algebra and trigonometry or any other branch of mathematics?
(v) Why is Calculus more difficult to absorb than algebra or trigonometry?
(vi) Are there any problems faced in our day-to-day life that can be solved more easily by Calculus than by arithmetic or algebra?
(vii) Are there any problems which cannot be solved without Calculus?
(viii) Why study Calculus at all?
(ix) Is Calculus different from other branches of mathematics?
(x) What type(s) of problems are handled by Calculus?

Introduction to Integral Calculus (Systematic Studies with Engineering Applications for Beginners) written by Rohde and GC Jain cover the following topics.






  • 1. Antiderivative(s) [or Indefinite Integral(s)]
    1.1 Introduction
    1.2 Useful Symbols, Terms, and Phrases Frequently Needed
    1.3 Table(s) of Derivatives and their corresponding Integrals
    1.4 Integration of Certain Combinations of Functions
    1.5 Comparison Between the Operations of Differentiation and Integration

  • 2. Integration Using Trigonometric Identities
    2.1 Introduction
    2.2 Some Important Integrals Involving sin x and cos x
    2.3 Integrals of the Form Ððdx=ða sin x þ b cos xÞÞ, where a, b 2 r

  • 3a. Integration by Substitution: Change of Variable of Integration
    3a.1 Introduction
    3a.2 Generalized Power Rule
    3a.3 Theorem
    3a.4 To Evaluate Integrals of the Form ð a sin x þ b cos x c sin x þ d cos x dx; where a, b, c, and d are constant

  • 3b. Further Integration by Substitution: Additional Standard Integrals
    3b.1 Introduction
    3b.2 Special Cases of Integrals and Proof for Standard Integrals
    3b.3 Some New Integrals
    3b.4 Four More Standard Integrals

  • 4a. Integration by Parts
    4a.1 Introduction
    4a.2 Obtaining the Rule for Integration by Parts
    4a.3 Helpful Pictures Connecting Inverse Trigonometric Functions with Ordinary Trigonometric Functions
    4a.4 Rule for Proper Choice of First Function

  • 4b. Further Integration by Parts: Where the Given Integral Reappears on Right-Hand Side
    4b.1 Introduction
    4b.2 An Important Result: A Corollary to Integration by Parts
    4b.3 Application of the Corollary to Integration by Parts to Integrals that cannot be Solved Otherwise
    4b.4 Simpler Method(s) for Evaluating Standard Integrals
    4b.5 To Evaluate

  • 5. Preparation for the Definite Integral: The Concept of Area
    5.1 Introduction
    5.2 Preparation for the Definite Integral
    5.3 The Definite Integral as an Area
    5.4 Definition of Area in Terms of the Definite Integral
    5.5 Riemann Sums and the Analytical Definition of the Definite Integral

  • 6a. The Fundamental Theorems of Calculus
    6a.1 Introduction
    6a.2 Definite Integrals
    6a.3 The Area of Function A(x)
    6a.4 Statement and Proof of the Second Fundamental Theorem of Calculus
    6a.5 Differentiating a Definite Integral with Respect to a Variable Upper Limit

  • 6b. The Integral Function Ð x11t dt, (x > 0) Identified as ln x or loge x
    6b.1 Introduction
    6b.2 Definition of Natural Logarithmic Function
    6b.3 The Calculus of ln x
    6b.4 The Graph of the Natural Logarithmic Function ln x
    6b.5 The Natural Exponential Function [exp(x) or ex]

  • 7a. Methods for Evaluating Definite Integrals
    7a.1 Introduction 197
    7a.2 The Rule for Evaluating Definite Integrals
    7a.3 Some Rules (Theorems) for Evaluation of Definite Integrals
    7a.4 Method of Integration by Parts in Definite Integrals
    7b Some Important Properties of Definite Integrals
    7b.1 Introduction
    7b.2 Some Important Properties of Definite Integrals
    7b.3 Proof of Property (P0)
    7b.4 Proof of Property (P5)
    7b.5 Definite Integrals: Types of Functions

  • 8a. Applying the Definite Integral to Compute the Area of a Plane Figure
    8a.1 Introduction
    8a.2 Computing the Area of a Plane Region
    8a.3 Constructing the Rough Sketch [Cartesian Curves]
    8a.4 Computing the Area of a Circle (Developing Simpler Techniques)

  • 8b. To Find Length(s) of Arc(s) of Curve(s), the Volume(s) of Solid(s) of Revolution, and the Area(s) of Surface(s) of Solid(s) of Revolution
    8b.1 Introduction
    8b.2 Methods of Integration
    8b.3 Equation for the Length of a Curve in Polar Coordinates
    8b.4 Solids of Revolution
    8b.5 Formula for the Volume of a “Solid of Revolution”
    8b.6 Area(s) of Surface(s) of Revolution

  • 9a. Differential Equations: Related Concepts and Terminology
    9a.1 Introduction
    9a.2 Important Formal Applications of Differentials (dy and dx)
    9a.3 Independent Arbitrary Constants (or Essential Arbitrary Constants)
    9a.4 Definition: Integral Curve
    9a.5 Formation of a Differential Equation from a Given Relation, Involving Variables and the Essential Arbitrary Constants (or Parameters)
    9a.6 General Procedure for Eliminating “Two” Independent Arbitrary Constants (Using the Concept of Determinant)
    9a.7 The Simplest Type of Differential Equations

  • 9b. Methods of Solving Ordinary Differential Equations of the First Order and of the First Degree
    9b.1 Introduction
    9b.2 Methods of Solving Differential Equations
    9b.3 Linear Differential Equations
    9b.4 Type III: Exact Differential Equations
    9b.5 Applications of Differential Equations


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