Back To Course

Math 104: Calculus14 chapters | 114 lessons | 9 flashcard sets

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Paul Bautista*

Ever feel like you are going around in circles? Like, periodically you have your ups and downs? Well, sines and cosines go up and down regularly too. In this lesson, learn how to integrate these circular functions.

Remember that you can calculate the definite integral of *f(x)* from *a* to *b* as being the anti-derivative of *f(x)* evaluated at *b* minus the anti-derivative of *f(x)* evaluated at *a*, which we write as the anti-derivative of *f(x)* from *a* to *b*. If you have an indefinite integral that is without limits, your indefinite integral is equal to the anti-derivative of *f(x)* plus some constant of integration *C*.

Let's take a look at trig functions, like *f(x)* = sin(*x*). If you recall, the derivative of sin(*x*) = cos(*x*), because the derivative is the slope of the tangent of the function. So here's sin(*x*) at *x*=0. The tangent has a slope of 1, which is the value of cos(*x*) evaluated at *x*=1. At *x* = *pi*/2, the slope of sin(*x*) is equal to 0. The value of cos(*x*) is equal to 0, because the derivative of sin(*x*) is equal to cos(*x*).

On the other side, the derivative of cos(*x*) with respect to *x* is equal to -sin(*x*). I can use these derivatives to determine what the integral, say of sin(*x*), is. The integral of sin(*x*) *dx* is equal to -cos(*x*) + *C*. How do you see this? Well, if I take the derivative of -cos(*x*) + *C*, I get minus the derivative of cos(*x*) plus the derivative of *C*. The derivative of a constant is zero because the slope of a line that has a constant value is zero, and the derivative of cos(*x*) is -sin(*x*). So my term *d/dx*(cos(*x*)) becomes - -sin(*x*), or just sin(*x*). So -cos(*x*) + *C* is an anti-derivative of sin(*x*). That is, if I take the derivative of -cos(*x*) + *C*, I end up with sin(*x*). You can make a similar argument for the integral of cos(*x*). Here, the integral of cos(*x*) is equal to sin(*x*) + *C*. You can see this by taking the derivative of sin(*x*) + *C*. That's just equal to cos(*x*).

There are a lot of trig functions out there, but really there are only two that you need to know the integral of off the top of your head, and those are sin(*x*) and cos(*x*). All of these other guys you'll generally look up in a table or you can determine just by knowing sin(*x*) and cos(*x*). So remember that **the integral of sin( x)dx = -cos(x) + C**, and the

So let's do an example. Let's say we want to integrate the function *f(x)* = sin(*x*) between *x*=0 and *x*=2*pi*. Remember that the integral is equal to the area under the curve. If you have a curve above the *x*-axis, that area is positive. But if you have something below the *x*-axis, this is actually a negative integral. So what you're really doing is adding this positive area and subtracting this negative area to find the integral. Just using your intuition, you know that if you're trying to find the integral - that is, this positive area minus this negative area - it might be zero, but let's see if we can calculate that exactly.

So let's calculate the integral from 0 to 2*pi* of sin(*x*)*dx*. Using the fundamental theorem, I know that's equal to the anti-derivative of sin(*x*) evaluated from 0 to 2*pi*. I also know that an anti-derivative is -cos(*x*), because the integral of sin(*x*)*dx* equals -cos(*x*) plus a constant. So let's plug in my anti-derivative, which is -cos(*x*). I've got -cos(*x*) evaluated from 0 to 2*pi*. Remember, this is like saying I'm evaluating this at 2*pi* and subtracting from it my evaluation of this at 0. So I've got -cos(2*pi*) - (-cos(0)). That's like -1 - -1, which is -1 + 1, which is just 0. Indeed, the integral from 0 to 2*pi* of sin(*x*) is 0; there's an equal amount of area above and below the *x*-axis.

What about a function like cos(*x*) + 1 between *x*=0 and *pi*? Again, let's use the fundamental theorem, and let's say that the integral from 0 to *pi* of (cos(*x*) + 1)*dx* is equal to the anti-derivative of this function evaluated from 0 to *pi*. Now let's break this integral up into two separate integrals so it's equal to the integral from 0 to *pi* of cos(*x*) plus the integral from 0 to *pi* of 1*dx*. This first term, the integral from 0 to *pi* of (cos(*x*))*dx*, is equal to the anti-derivative from 0 to *pi*. That anti-derivative is sin(*x*), so we plug that in here and I get sin(*x*) evaluated from 0 to *pi*, so that's sin(*pi*)-sin(0). Well, that's just equal to 0.

Okay, what about the second term, 0 to *pi* *dx*? 0 to *pi* *dx* is equal to the anti-derivative evaluated from 0 to *pi*. The anti-derivative of 1 is *x* + *C*, and remember we're ignoring *C* here because we're looking at a definite integral. So I'm going to plug *x* in for my anti-derivative and evaluate from 0 to *pi*, and I get *pi* - 0, which is just *pi*. So my total integral is equal to 0 + *pi*, and that's just *pi*, so the area under the curve here is equal to *pi*. That is, the integral from 0 to *pi* of cos(*x*) + 1 is equal to *pi*.

Let's review. There are a lot of trig functions, but really you just need to memorize two anti-derivatives. That is, you need to know the integral of sin(*x*)*dx* is equal to -cos(*x*) + *C*. That's because if you take the derivative of -cos(*x*) + *C* you get back sin(*x*). The integral of cos(*x*)*dx* is equal to sin(*x*) + *C*. And again, that's because if you take the derivative of sin(*x*) + *C* you end up getting back cos(*x*).

To unlock this lesson you must be a Study.com Member.

Create your account

Already a member? Log In

BackDid you know… We have over 49 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Back To Course

Math 104: Calculus14 chapters | 114 lessons | 9 flashcard sets

- Go to Continuity

- Go to Limits

- Calculating Integrals of Simple Shapes 7:50
- Anti-Derivatives: Calculating Indefinite Integrals of Polynomials 11:55
- How to Calculate Integrals of Trigonometric Functions 8:04
- How to Solve Integrals Using Substitution 10:52
- Substitution Techniques for Difficult Integrals 10:59
- Using Integration By Parts 12:24
- Partial Fractions: How to Factorize Fractions with Quadratic Denominators 12:37
- How to Integrate Functions With Partial Fractions 9:11
- Understanding Trigonometric Substitution 10:29
- How to Use Trigonometric Substitution to Solve Integrals 13:28
- How to Solve Improper Integrals 11:01
- Go to Integration and Integration Techniques

- GATE Exam - EY (Ecology & Evolution): Test Prep & Syllabus
- GATE Exam - GG (Geology & Geophysics): Test Prep & Syllabus
- ISC Physics: Study Guide & Syllabus
- BITSAT Exam - Physics: Study Guide & Test Prep
- BITSAT Exam - English & Logic: Study Guide & Test Prep
- Studying for Economics 101
- Conduction of Electricity
- Structural Geology
- Radioactive Prospecting Methods
- Seismic Methods of Prospecting
- Roots of the Vietnam War: Learning Objectives & Activities
- Unrest in Vietnam During the Eisenhower Years: Learning Objectives & Activities
- John F. Kennedy and the Vietnam War: Learning Objectives & Activities
- Vietnam War During the Nixon Years: Learning Objectives & Activities
- Major Battles & Offensives of the Vietnam War: Learning Objectives & Activities
- The Vietnam War After American Involvement: Learning Objectives & Activities
- Lyndon B. Johnson and the Vietnam War: Learning Objectives & Activities

- What Is the Northwest Passage? - Explorers, Definition & History
- How to Calculate the Break Even Point - Definition & Formula
- Types of Altruism in Psychology
- Parts of a Leaf: Lesson for Kids
- Informal Assessment Ideas for Social Studies
- Understanding Parallelogram of Forces
- Andar Conjugation: Preterite & Future Tense
- Formative Assessment Ideas for Social Studies
- Quiz & Worksheet - Macbeth as Tragic Hero
- Quiz & Worksheet - Absolute Zero
- Quiz & Worksheet - What Is Revenue in Accounting?
- Quiz & Worksheet - Trace Evidence Examination
- Quiz & Worksheet - Transition Statements in Writing
- Orchestra Instruments List & Flashcards
- Articles of Confederation Flashcards

- MTEL Business: Practice & Study Guide
- High School Trigonometry: Homeschool Curriculum
- Remedial 11th Grade English
- Algebra II: High School
- Romeo and Juliet by Shakespeare: Study Guide
- AEPA: Effects of Population on the Environment
- Living Organisms & Ecosystems
- Quiz & Worksheet - Weak Acids, Weak Bases, and Buffers
- Quiz & Worksheet - The Life Course Perspective
- Quiz & Worksheet - Characteristics of Anthropology
- Quiz & Worksheet - Types of Muscle Tissue
- Quiz & Worksheet - Impact of Inflation and Fraud on Older Adults

- The Different Factors Affecting Personality
- Reactive Hyperemia: Definition & Test
- Erosion Lesson Plan
- Best MCAT Prep Course
- Figurative Language Lesson Plan
- PSAT Tips & Tricks
- The Masque of the Red Death Lesson Plan
- Can You Use a Calculator on the GMAT?
- Adult Learning Programs of Alaska
- Meiosis Lesson Plan
- Silk Road Lesson Plan
- SAT Subject Tests: Registration & Test Dates

Browse by subject