Integral Calculus Problem Set III
Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE ALevel; 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!
Calculus  Integral Calculus Problem Set III  Outline of Contents:
Examples and solved problems  Reduction formulas, improper integrals, reducing the integrand to partial fractions, more of definite integrals.
Here are the kind of cases we will cover in this tutorial.
1
∫ log x dx  Solve this Integral. We eventually simplify this to the limiting value of [−h log h − (1 − h)] as h>0
0
Showing that the area bound by the curves e^{a}^{x} sin(bx) and e^{a}^{x} cos bx, and the positive xaxis is b/(a^{2} + b^{2}) and a/(a^{2} + b^{2}) respectively. We solve these using integration by parts, and then using limiting values (as x>infinity)
Establish a reduction formula for ∫ x^{n} e^{ax} dx and apply it to evaluate ∫ x^{3} e^{ax} dx. We solve this using integration by parts, to arrive at a reduction formula, and then put n = 1, 2, 3 respectively
Obtain a reduction formula for ∫ x^{m} sin nx dx. Apply integration of parts. You get a reduction formula including a term of ∫ x^{m1} cos nx dx, on which you again apply integration of parts to get a formula including the term ∫ x^{m2} sin nx dx.
Obtain a reduction formula for ∫ x^{n} e^{−x} dx and show that the value equals n! when we compute the value of the definite integral between x = 0 and infinity.
We solve this using integration by parts. One term involves a multiple of ∫ x^{n1} e^{−x} dx and the other one can be eliminated altogeter because it tends to 0
Obtain a reduction formula for ∫ x^{m} (log x)^{n} dx and use it to evaluate
1
∫ x^{4} (log x)^{3} dx.
0
We also pick up other interesting problems involving polynomials in the denominator. These are solved using partial fractions.
Quick and introductory definitions related to Funtions, Limits and Continuity Functions, Limits and Continuity  Solved Problem Set I  The Domain, Range, Plots and Graphs of Functions; L'Hospital's 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 Functions, Limits and Continuity  Solved Problem Set III  Continuity and Intermediate Value Theorems Introductory concepts and definitions related to Differentiation  Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc Differential Calculus  Solved Problem Set I  Common Exponential, Log , trigonometric and polynomial functions Differential Calculus  Solved Problem Set II  Derivability and continuity of functins  Change of Indepndent Variables  Finding Nth Derivatives 
Differential Calculus  Solved Problems Set III Maximia, Minima, Extreme Values, Rolle's Theorem Differential Calculus  Solved Problems Set IV  Points of Inflexion, Radius of Curvature, Curve Sketching Differential Calculus  Solved Problems Set V  Curve Sketching, Parametric Curves 
Introducing Integral Calculus  Definite and Indefinite Integrals  using Substitution , Integration By Parts, ILATE rule Integral Calculus  Solved Problems Set I  Basic examples of polynomials and trigonometric functions, area under curves Integral Calculus  Solved Problems Set II  More integrals, functions involving trigonometric and inverse trigonometric ratios Integral Calculus  Solved Problems Set III  Reduction Formulas, Using Partial FractionsI Integral Calculus  Solved Problems Set IV  More of integration using partial fractions, more complex substitutions and transformations Integral Calculus  Solved Problems Set V Integration as 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 Differential Equations  Solved Problems  Set II  D operator, auxillary equation, General Solution Differential Equations  Solved Problems  Set III  More Differential Equations Differential Equations  Solved Problems  Set IV

