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#### The Fundamental Theorem of Calculus

Three Different Quantities
The Whole as Sum of Partial Changes
The Indefinite Integral as Antiderivative
The FTC and the Chain Rule

#### 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

Substitution for Indefinite Integrals
Revised Table of Integrals
Substitution for Definite Integrals

#### Area Between Curves

The Slice and Dice Principle
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary

#### Volumes

Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
Volumes Rotating About the $y$-axis

Behind IBP
Examples
Going in Circles

#### Integrals of Trig Functions

Basic Trig Functions
Product of Sines and Cosines (1)
Product of Sines and Cosines (2)
Product of Secants and Tangents
Other Cases

#### Trig Substitutions

How it works
Examples
Completing the Square

#### Partial Fractions

Introduction
Linear Factors
Improper Rational Functions and Long Division
Summary

#### Strategies of Integration

Substitution
Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

#### Improper Integrals

Type I Integrals
Type II Integrals
Comparison Tests for Convergence

#### Differential Equations

Introduction
Separable Equations
Mixing and Dilution

#### Models of Growth

Exponential Growth and Decay
Logistic Growth

#### Infinite Sequences

Close is Good Enough (revisited)
Examples
Limit Laws for Sequences
Monotonic Convergence

#### Infinite Series

Introduction
Geometric Series
Limit Laws for Series
Telescoping Sums and the FTC

#### Integral Test

The Integral Test
When the Integral Diverges
When the Integral Converges

#### 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
Rearrangements

The Ratio Test
The Root Test
Examples

#### Strategies for testing Series

List of Major Convergence Tests
Examples

#### Power Series

Finding the Interval of Convergence
Other Power Series

#### 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

What are Taylor Polynomials?
How Accurate are Taylor Polynomials?
What can go Wrong?
Other Uses of Taylor Polynomials

#### Partial Derivatives

Definitions and Rules
The Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions

#### Multiple Integrals

Background
What is a Double Integral?
Volumes as Double Integrals

#### Iterated Integrals over Rectangles

One Variable at the Time
Fubini's Theorem
Notation and Order

#### Double Integrals over General Regions

Type I and Type II regions
Examples
Order of Integration
Area and Volume Revisited

### Other Cases

There are a number of other trig functions that you might want to integrate. Here is a sketch of how to handle them. Some of these techniques are fairly advanced, so check with your instructor to see whether they are on the syllabus for your section.

• $\int \sec^n(x) \tan^m(x)\, dx$ with $n$ odd and $m$ even.

First convert all of the tangents into secants, so that we have something of the form $\int \sec^n(x)\, dx$. Then do an integration by parts with $u=\sec^{n-2}(x)$ and $dv = \sec^2(x)\, dx$. After a little algebra, you get a recursive formula for $\int \sec^{n}(x)\, dx$ in terms of $\int \sec^{n-2}(x) \,dx$. Repeat the process until you get it down to $\int \sec(x)\,dx = \ln\lvert\sec(x)+\tan(x)\rvert+C$.

 Example 1: $\int \sec^3(x)\, dx$. Integrate by parts with $u=\sec(x)$, $dv=\sec^2(x)$, so $du=\sec(x)\tan(x)\, dx$ and $v=\tan(x)$:$$\int \sec^3(x) \,dx = \sec(x)\tan(x) - \int \sec(x)\tan^2(x) \,dx$$Then, since $\tan^2(x)=\sec^2(x)-1$, we can rewrite this as $$\int \sec^3(x) \,dx = \sec(x)\tan(x) + \int \sec(x) \,dx - \int \sec^3(x) \,dx.$$Adding $\int \sec^3(x)\, dx$ to each side and dividing by 2 gives$$\int \sec^3(x) \,dx = \frac{\sec(x)\tan(x) + \int \sec(x) \,dx}{2} = \frac{\sec(x)\tan(x) + \ln\lvert\sec(x)+\tan(x)\rvert}{2}+C.$$

• Integrals of the form $\int\csc^n(x)\cot^m(x)\,dx$.

These follow the same strategy as $\int \sec^n(x)\tan^m(x)\, dx$, with $\csc$ replacing $\sec$ and $\cot$ replacing $\tan$. The only difference is with signs. Remember that the derivative of $\cot(x)$ is minus $\csc^2(x)$, and that the derivative of $\csc(x)$ is minus $\csc(x)\cot(x)$.

• Other products and ratios of trig functions.

These can all be converted to the previous cases by writing everything in terms of sines and cosines and using the identity $\sin^2(x)+\cos^2(x)=1$.

 Example 2: Since $$\sec^2(x)\csc^2(x) = \frac{1}{\sin^2(x)\cos^2(x)} = \frac{\sin^2(x)+\cos^2(x)}{\sin^2(x)\cos^2(x)}$$$$\qquad\qquad\qquad=\frac{1}{\cos^2(x)}+ \frac{1}{\sin^2(x)}=\sec^2(x) + \csc^2(x),$$ we have $$\int \sec^2(x)\csc^2(x)\, dx = \tan(x) - \cot(x) + C.$$

 Example 3: Since $$\sin(x) \tan(x) = \frac{\sin^2(x)}{\cos(x)} = \frac{1-\cos^2(x)}{\cos(x)} = \sec(x) - \cos(x),$$ we have $$\int \sin(x)\tan(x)\, dx = \int \sec(x)\, dx - \int \cos(x) \,dx = \ln\bigl\lvert\sec(x)+\tan(x)\bigr\rvert - \sin(x) +C.$$

• Products of trig functions with different frequencies, like $\sin(x)\cos(2x)$ or $\cos(2x)\cos(3x)$, can be handled either with integration-by-parts (going in circles) or by using the addition-of-angle formulas:
\begin{eqnarray*}\sin(A)\sin(B)&=&\frac{\cos(A-B)-\cos(A+B)}{2}\cr \cos(A)\cos(B)&=&\frac{\cos(A-B)+\cos(A+B)}{2}\cr\sin(A)\cos(B)&=&\frac{\sin(A+B)+\sin(A-B)}{2}\end{eqnarray*}

 Example 4: We compute $\int \sin(x)\cos(2x)\,dx$ in two ways. By parts: $$\int \sin(x)\cos(2x) dx \overset{\fbox{IBP:  u\,=\,\cos(2x),\\\quad dv\,=\,\sin(x)\,dx}}{=} -\cos(x)\cos(2x) - \int 2\cos(x)\sin(2x)\,dx$$ $$\int \sin(x)\cos(2x) \,dx \overset{\fbox{IBP: u\,=\,2\sin(2x),\\ \quad dv\,=\,\cos(x)\,dx}}{=} -\cos(x)\cos(2x) - \left(2\sin(x)\sin(2x) -4\int \sin(x)\cos(2x) \,dx\right)$$ With some algebra, we find $$-3\int\sin(x)\cos(2x)\,dx = -\cos(x)\cos(2x) - 2 \sin(x)\sin(2x) +C,$$ i.e., $$\int\sin(x)\cos(2x)\,dx= \frac{\cos(x)\cos(2x)+2\sin(x)\sin(2x)}{3}+C$$ Or, using addition-of-angle formulas: \begin{eqnarray*}\int \sin(x) \cos(2x) dx = & \int \frac{\sin(3x)+\sin(-x)}{2} dx & \quad \hbox{(using the formula for $\sin(A)\cos(B)$)}\cr =&\int \frac{\sin(3x)-\sin(x)}{2} dx & \quad \hbox{(since $\sin(-x)=-\sin(x)$)}\cr =& \frac{-\cos(3x)}{6} + \frac{\cos(x)}{2} + C\end{eqnarray*} The two answers don't look the same, but in fact they are equal, as you can check by applying the addition-of-angle formulas to the first answer.