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The Fundamental Theorem of Calculus

Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1

The Indefinite Integral and the Net Change

Indefinite Integrals and Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy


Substitution for Indefinite Integrals
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals

Area Between Curves

Computation Using Integration
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing


Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice

Integration by Parts

Integration by Parts
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade

Integrals of Trig Functions

Antiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (only even powers)
Product of Secants and Tangents
Other Cases

Trig Substitutions

How Trig Substitution Works
Summary of trig substitution options
Completing the Square

Partial Fractions

Linear Factors
Irreducible Quadratic Factors
Improper Rational Functions and Long Division

Strategies of Integration

Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

Type 1 - Improper Integrals with Infinite Intervals of Integration
Type 2 - Improper Integrals with Discontinuous Integrands
Comparison Tests for Convergence

Differential Equations

Separable Equations
Mixing and Dilution

Models of Growth

Exponential Growth and Decay
Logistic Growth

Infinite Sequences

Approximate Versus Exact Answers
Examples of Infinite Sequences
Limit Laws for Sequences
Theorems for and Examples of Computing Limits of Sequences
Monotonic Covergence

Infinite Series

Geometric Series
Limit Laws for Series
Test for Divergence and Other Theorems
Telescoping Sums and the FTC

Integral Test

Road Map
The Integral Test
Estimates of Value of the Series

Comparison Tests

The Basic Comparison Test
The Limit Comparison Test

Convergence of Series with Negative Terms

Introduction, Alternating Series,and the AS Test
Absolute Convergence

The Ratio and Root Tests

The Ratio Test
The Root Test

Strategies for testing Series

Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2

Power Series

Radius and Interval of Convergence
Finding the Interval of Convergence
Power Series Centered at $x=a$

Representing Functions as Power Series

Functions as Power Series
Derivatives and Integrals of Power Series
Applications and Examples

Taylor and Maclaurin Series

The Formula for Taylor Series
Taylor Series for Common Functions
Adding, Multiplying, and Dividing Power Series
Miscellaneous Useful Facts

Applications of Taylor Polynomials

Taylor Polynomials
When Functions Are Equal to Their Taylor Series
When a Function Does Not Equal Its Taylor Series
Other Uses of Taylor Polynomials

Partial Derivatives

Visualizing Functions in 3 Dimensions
Definitions and Examples
An Example from DNA
Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions

Multiple Integrals

What is a Double Integral?
Volumes as Double Integrals

Iterated Integrals over Rectangles

How To Compute Iterated Integrals
Examples of Iterated Integrals
Cavalieri's Principle
Fubini's Theorem
Summary and an Important Example

Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Order of Integration

Substitution for Indefinite Integrals

Integration by substitution, or $u$-substitution, is the most common technique of finding an antiderivative.  It allows us to find the antiderivative of a function by reversing the chain rule.  To see how it works, consider the following example.

Let $f(x)=(x^2-2)^8$.  Then  $f'(x)=8(x^2-2)^7(2x)$ by the chain rule.  Remember that we have the "inside part" $(x^2 -2)$, and its derivative $(2x)$.  We can use this knowledge to antidifferentiate, if we have an integrand that has an "inside part" and is multiplied by the derivative of that "inside part".

To see this, consider $\int 8(x^2-2)^7(2x)\,dx$.  We make a substitution to make it clear what to do:  Let $u$ be the inside part, so $u=x^2-2$.  We differentiate $u$, and use the notation $du=2x\,dx$ (instead of $\frac{du}{dx}=2x$; you will see why when we subsitute back into the integrand).  Then $\int 8(x^2-2)^7(2x)\,dx=\int 8u^7\,du$ by directly replacing every element in the first integrand, including the $dx$, by exactly what it becomes via the substution (in blue above)This integral is now easy to evaluate, giving $u^8+c=(x^2-2)^8+c$.

Notice our integrand above was a composite function, $f(g(x))$ where $g(x)=x^2-2$ was the inside part.  This $u$-substitution process can be stated formally as shown below.  Notice that the first integrand is a composite function multiplied by the derivative of the inside part.

$$ \int f(g(x))g'(x)\,dx = \int f(u)\,du.$$

Another example:

$\displaystyle \int \cos(x^3)\cdot 3x^2\,dx\overset{\fbox{$ \,\,u\,=\,x^3,\\ du\,=\,3x^2\,dx$}\\}{=}\int \cos(u)\,du= \sin(u)+c=\sin(x^3)+c$.    Do:  Differentiate to check this answer!

By choosing a suitable function $u$, we can often convert hard integrals into much easier integrals that we know how to evaluate. Unfortunately, there is no magic formula for deciding what $u$ should be — this is a toolkit, not a recipe.  Some examples are in the video that follows.