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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 Antiderivatives
A Table of Common Antiderivatives
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
Integration by Parts
Behind IBP
Examples
Going in Circles
Tricks of the Trade
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
Quadratic 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
Road Map
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 and Root Tests
The Ratio Test
The Root Test
Examples
Strategies for testing Series
List of Major Convergence Tests
Examples
Power Series
Radius and Interval of Convergence
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 $u=\sec^{n2}(x)$ and $dv = \sec^2(x)\, dx$. After a little algebra, you get a recursive formula for $\int \sec^{n}(x)\, dx$ in terms of $\int \sec^{n2}(x) \,dx$. 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 integrationbyparts (going in circles) or by using the additionofangle formulas:
\begin{eqnarray*}\sin(A)\sin(B)&=&\frac{\cos(AB)\cos(A+B)}{2}\cr \cos(A)\cos(B)&=&\frac{\cos(AB)+\cos(A+B)}{2}\cr\sin(A)\cos(B)&=&\frac{\sin(A+B)+\sin(AB)}{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 additionofangle 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 additionofangle formulas to the first answer.

