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 x^{2} = y, in case 3 we make the substitution x^{2} + 1 = y and in case 4 we make the substitution x = tan θ.
These are the functions we will integrate :
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Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE/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 Nth Derivatives  Examples and solved problems  related to derivability and continuity of functions; changing the independent variable in a differential equation; finding the Nth 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 NonLinear 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. 