 Recommended Books for Mathematics and Calculus Lovers:
--------xxx-----------  Calculus - Integral Calculus Problem Set IV - Outline of Contents:Here's a quick look at the examples and solved problems which we will learn to solve in this tutorial - More of integrals involving decomposition into partial fractions, more complex substitutions and transformationsThese are the functions which we will integrate (with respect to dx ). Partial Fractions is one of the topics which we pay special attention to, in this tutorial. Apart from using partial fractions, we also show how interesting substitutions can simplify our task. For example in case 2 we make the substitution x2 = y, in case 3 we make the substitution x2 + 1 = y and in case 4 we make the substitution x = tan θ.
These are the functions we will integrate :
|
Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!
| Quick and introductory definitions related to Funtions, Limits and Continuity - Defining the domain and range of a function, the meaning of continuity, limits, left and right hand limits, properties of limits and the "lim" operator; some common limits; defining the L'Hospital rule, intermediate and extreme value theorems. | Functions, Limits and Continuity - Solved Problem Set I - The Domain, Range, Plots and Graphs of Functions; L'Hospital's Rule- - Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule. |
Functions, Limits and Continuity - Solved Problem Set II - Conditions for Continuity, More Limits, Approximations for ln (1+x) and sin x for infinitesimal values of x - More advanced cases of evaluating limits, conditions for continuity of functions, common approximations used while evaluating limits for ln ( 1 + x ), sin (x); continuity related problems for more advanced functions than the ones in the first group of problems (in the last tutorial). | Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems - Problems related to Continuity, intermediate value theorem. |
| Introductory concepts and definitions related to Differentiation - Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc - Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem, Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation. |
Differential Calculus - Solved Problem Set I - Common Exponential, Log , trigonometric and polynomial functions - Examples and solved problems - differentiation of common algebraic, exponential, logarithmic, trigonometric and polynomial functions and terms; problems related to differentiability . | Differential Calculus - Solved Problem Set II - Derivability and continuity of functins - Change of Indepndent Variables - Finding N-th Derivatives - Examples and solved problems - related to derivability and continuity of functions; changing the independent variable in a differential equation; finding the N-th derivative of functions |
Differential Calculus - Solved Problems Set III- Maximia, Minima, Extreme Values, Rolle's Theorem - Examples and solved problems - related to increasing and decreasing functions; maxima, minima and extreme values; Rolle's Theorem |
| Differential Calculus - Solved Problems Set IV - Points of Inflexion, Radius of Curvature, Curve Sketching - Examples and solved problems - Slope of tangents to a curve, points of inflexion, convexity and concavity of curves, radius of curvature and asymptotes of curves, sketching curves |
Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves - More examples of investigating and sketching curves, parametric representation of curves |
Introducing Integral Calculus - Definite and Indefinite Integrals - using Substitution , Integration By Parts, ILATE rule - Theory and definitions. What integration means, the integral and the integrand. Indefinite integrals, integrals of common functions. Definite integration and properties of definite integrals; Integration by substitution, integration by parts, the LIATE rule, Integral as the limit of a sum. Important forms encountered in integration. | Integral Calculus - Solved Problems Set I - Basic examples of polynomials and trigonometric functions, area under curves - Examples and solved problems - elementary examples of integration involving trigonometric functions, polynomials; integration by parts; area under curves. |
| Integral Calculus - Solved Problems Set II - More integrals, functions involving trigonometric and inverse trigonometric ratios - Examples and solved problems - integration by substitution, definite integrals, integration involving trigonometric and inverse trigonometric ratios. | Integral Calculus - Solved Problems Set III - Reduction Formulas, Using Partial FractionsI- Examples and solved problems - Reduction formulas, reducing the integrand to partial fractions, more of definite integrals |
Integral Calculus - Solved Problems Set IV - More of integration using partial fractions, more complex substitutions and transformations - Examples and solved problems - More of integrals involving partial fractions, more complex substitutions and transformations | Integral Calculus - Solved Problems Set V- Integration as a summation of a series - Examples and solved problems - More complex examples of integration, examples of integration as the limit of a summation of a series |
| Introduction to Differential Equations and Solved Problems - Set I - Order and Degree, Linear and Non-Linear Differential Equations, Homogeneous Equations, Integrating Factor - Theory and definitions. What a differential equation is; ordinary and partial differential equations; order and degree of a differential equation; linear and non linear differential equations; General, particular and singular solutions; Initial and boundary value problems; Linear independence and dependence; Homogeneous equations; First order differential equations; Characteristic and auxiliary equations. Introductory problems demonstrating these concepts. Introducing the concept of Integrating Factor (IF). |
Differential Equations - Solved Problems - Set II - D operator, auxillary equation, General Solution - Examples and solved problems - Solving linear differential equations, the D operator, auxiliary equations. Finding the general solution ( CF + PI ) |
Differential Equations - Solved Problems - Set III - More Differential Equations - More complex cases of differential equations. | Differential Equations - Solved Problems - Set IV - Still more differential equations. |