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Antiderivatives From Derivatives of Basic Functions


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Lecture Notes

Antidifferentiation

Document PDF
Section 2, Page 2

Antiderivatives and indefinite integrals are defined. Constants of integration and integrands are also defined.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Concept of derivative (section 2 of lecture 1)
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Antidifferentiation: Guess-and-Check

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Section 3, Page 2 to page 3

Guess-and-check method for finding antiderivatives. Includes an example and some helpful rules.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: Antidifferentiation (section 2 of this lecture)
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Online Textbook Chapters

Introduction to Anti-Derivatives

Document Document

Definition of the indefinite integral or anti-derivative and its use in finding information about a function when its derivative is known.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: None
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Non-Uniqueness of Anti-Derivatives

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Discussion of the fact that any constant can be added to an antiderivative without changing the validity of that antiderivative.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Anti-derivatives (OT19.1)
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Anti-Derivatives by Inspection

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Applying differentiation rules backwards to find anti-derivatives. A list of types of functions that can or cannot be anti-differentiated. Linearity of the operation of anti-differentiation.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Anti-derivatives (OT19.1)
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Basic Techniques for Integrals

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Rules for integrating polynomials and other simple integrals by inspection, as well as techniques for integrating by substitution, parts, and partial fractions.

Course: 18.013A Calculus with Applications, Spring 2005
Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Anti-derivatives (OT19.1)
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Exam Questions

Evaluating Definite Integrals

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Problem 4 (page 6) to problem (page 7)

Five-part problem evaluating integrals involving the substitution method, logarithmic functions, and trigonometric functions.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None

Solution (PDF) Pages 3 to 6

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Antiderivatives Using Polynomial Division

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Problem 2 (page 3)

Computing the antiderivative of a fraction of two polynomials.

Course: 18.01 Single Variable Calculus, Fall 2005
Instructor: Prof. Jason Starr
Prior Knowledge: None

Solution (PDF) Pages 1 to 2

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Indefinite Integrals

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Problem 9 (page 1)

Three integrals to be evaluated.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 2

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Indefinite Integrals

Document PDF
Problem 3 (page 1)

Three integrals to be evaluated.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 1

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Indefinite Integrals

Document PDF
Problem 5a (page 2)

Two integrals to be evaluated.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 2

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Trigonometric and Logarithmic Integrals

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Problem 1 (page 1)

Deriving a trigonometric formula and differentiating a logarithmic expression, then using those results to evaluate two integrals.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 1

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Definite Integrals

Document PDF
Problem 1 (page 1)

Two integrals to be evaluated.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF)# Page 1

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Trigonometric Formulas and Identities

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(page 1)

A list of trigonometric and inverse trigonometric identities and formulas involving integrals and derivatives.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None
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Differentiation Formulas: Polynomials, Products, Quotients

Document PDF - 2.2 MB
Problem 1E-1 (page 4) to problem 1E-5 (page 5)

Five questions which involve taking derivatives and antiderivatives of polynomials, finding the points on a graph which have horizontal tangent lines, and finding derivatives of rational functions.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF - 4.0 MB) Page 9

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Differentials and Indefinite Integration

Document PDF - 2.2 MB
Problem 3A-1 (page 21) to problem 3A-3 (page 21)

Three questions which involve evaluating five differentials and twenty indefinite integrals using a range of techniques.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF - 4.0 MB) Pages 37 to 39

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Inverse Trigonometric and Hyperbolic Functions

Document PDF - 2.2 MB
Problem 5A-1 (page 35) to problem 5A-6 (page 35)

Six questions which involve evaluating integrals and derivatives of these functions, as well as graphing them and finding tangent lines or average values.

Course: 18.01 Single Variable Calculus, Fall 2006
Instructor: Prof. David Jerison
Prior Knowledge: None

Solution (PDF - 4.0 MB) Pages 69 to 71

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Analysis of Graphs Limits of Functions Asymptotic & Unbounded Behavior Continuity: Property of Functions Parametric, Polar & Vector Function Concept of the Derivative Derivative at a Point Derivative as a Function Second Derivatives Applications of Derivatives Computation of Derivatives Interpretations & Properties of Definite Integrals Applications of Integrals Fundamental Theorem of Calculus Applications of Antidifferentiation Numerical Approximations to Definite Integ Concept of Series Series of Constants Taylor Series Graphs of Functions Intuitive Understanding: Limiting Process Calculating Limits Using Algebra Asymptotic Behavior: Infinity Continuity Continuity in Terms of Limits Analysis of Planar Curves Derivative Presented Graphically, Numerically & Analytically Derivative Interpreted as Instantaneous Rate of Change Derivative: Limit of the Difference Quotient Differentiability & Continuity Tangent Line - Curve at a Point & Local Linear Approximation Instantaneous Rate of Change Approximate Rate of Change Characteristics of f & f Relationship Between Behavior of f & Sign of f Mean Value Theorem & Geometric Consequences Characteristics: Graphs of f, f & f Relationship Between Concavity of f & Sign of f'' Points of Inflection - Concavity Changes Analysis of Curves Applications of Derivatives Optimization: Absolute & Relative Extrema Modeling Rates of Change Implicit Differentiation & Derivatives of Inverse Functions Derivative as a Rate of Change l'Hôpital's Rule Geometric Interpretation of Differential Equations Derivatives of Basic Functions Rules: Derivative of Sums, Products & Quotients of Functions Chain Rule & Implicit Differentiation Derivatives: Parametric, Polar & Vector Functions Definite Integral - Limit of Riemann Sums Rate of Change of a Quantity/Change of Quantity over Interval Properties of Definite Integrals Appropriate Integrals Evaluate Definite Integrals Representation of Specific Antiderivatives Antiderivatives From Derivatives of Basic Functions Integration by Substitution, Parts & Partial Fractions Improper Integrals Separable Equations & Modeling Riemann & Trapezoidal Sums Series, Convergence, Divergence Motivating Examples Geometric Series with Applications The Harmonic Series Alternating Series - Error Bound Series as Riemann Sums & the Integral Test Ratio Test - Convergence/Divergence Comparison Test Taylor Polynomial Approximation Maclaurin & Taylor Series Maclaurin Series for Basic Functions Manipulating Taylor Series Power Series Defining Functions Radius/Interval of Convergence Lagrange Error Bound