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 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


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
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
Completing the Square

Partial Fractions

Linear Factors
Quadratic Factors
Improper Rational Functions and Long Division

Strategies of Integration

Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

Type I Integrals
Type II Integrals
Comparison Tests for Convergence

Differential Equations

Separable Equations
Mixing and Dilution

Models of Growth

Exponential Growth and Decay
Logistic Growth

Infinite Sequences

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

Infinite Series

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

Alternating Series and the AS Test
Absolute Convergence

The Ratio and Root Tests

The Ratio Test
The Root Test

Strategies for testing Series

List of Major Convergence Tests

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

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
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.

    1. 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., $$ \int\sin(x)\cos(2x)\,dx= \frac{\cos(x)\cos(2x)+2\sin(x)\sin(2x)}{3}+C$$

    2. 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.