Integral Calculus- Problem Set III
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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!
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.
∫ log x dx -- Solve this Integral. We eventually simplify this to the limiting value of [−h log h − (1 − h)] as h->0
Showing that the area bound by the curves eax sin(bx) and eax cos bx, and the positive x-axis is b/(a2 + b2) and a/(a2 + b2) respectively.
We solve these using integration by parts, and then using limiting values (as x->infinity)
Establish a reduction formula for ∫ xn eax dx and apply it to evaluate ∫ x3 eax 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 ∫ xm sin nx dx.
Apply integration of parts. You get a reduction formula including a term of ∫ xm-1 cos nx dx, on which you again apply integration of parts to get a formula including the term ∫ xm-2 sin nx dx.
Obtain a reduction formula for ∫ xn 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 ∫ xn-1 e−x dx and the other one can be eliminated altogeter because it tends to 0
Obtain a reduction formula for ∫ xm (log x)n dx and use it to evaluate
∫ x4 (log x)3 dx.
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 N-th 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 Non-Linear 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