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:
*Kelly Sjol*

In this lesson, we use each of the common integration techniques to solve different integrals. It's not always obvious which technique will be the easiest, so being familiar with an arsenal of methods might save you a lot of work!

Let's take a minute to review some techniques for integration. Here we're going to integrate the **indefinite integral** - that means it has no limits - of *f(x)dx*. We know that the integral of *f(x)dx* equals the anti-derivative as a function of *x* plus a constant of integration. If you take the derivative of the anti-derivative, you get back your original function. So what are some of the ways we know to find integrals?

First, we can use a table. This may be a table in a book, online or what you have memorized. For example, the integral of *x*^2*dx* = (1/3)*x*^3 + *C*. You know this because you know how to integrate polynomials. The integral of sin(*x*)*dx* = -cos(*x*) + *C*. The integral of *e^(x)dx* = *e^(x)* + *C*. For each of these cases, if you take the derivative of the right side, you end up with the **integrand**. This is true of all integrals; it's how you calculate an integral.

The second way we know to calculate integrals is by substitution. In this case, we're going to take an integral that depends on *x*, and we're going to make a substitution where *u* equals some new function of *x*. By plugging in *u*, we hope to end up with a simpler integral that we can integrate with respect to *u*. For example, we have the integral sin(2*x*)*dx*. I want to substitute *u* for 2*x*, so *u*=2*x* and *du*=2*dx*. So I can plug those in, both for 2*x* and for *dx*, and my integral becomes 1/2 sin(*u*)*du*. I can use a table to solve this, because the integral of sin(*u*) is -cos(*u*). I get -1/2 cos(*u*) + *C*. Now I want to plug in 2*x* where I have *u* - that's my original substitution - so I get *x* back in my final answer. I get -1/2 cos(2*x*) + *C*. If I take the derivative of this, I end up with sin(2*x*). That's solving by substitution, and that is by far what you are going to use the most when solving integrals by hand, but there are a couple of other methods that you should be aware of.

One is integration by parts. Here you have the integral of *udv* = *uv* minus the integral of *vdu*. This is just a rearrangement of the product rule. An example would be the integral of *xe*^(*x*)*dx*. Here I'm going to set *x* equal to a new variable, *u* so that *du*=*dx*. I'm going to set *e*^(*x*)*dx* equal to *dv*, so *v* has to be equal to *e*^*x* because the derivative of *e*^*x* is *e*^(*x*)*dx*. If I plug *u*, *v*, *du* and *dv* into the right side of my equation for integration by parts, I end up with *xe*^*x* (that's *uv*) minus the integral of *e*^(*x*)*dx* (that's *vdu*). At this point, I can use a table in my head because I have memorized this integral. The integral of *e*^(*x*)*dx* is *e*^*x*, so if I plug that in, my integral becomes *xe*^*x* - *e*^*x* + *C*. If I take the derivative, I end up with *xe*^*x*.

The last way you can solve an integral is by Riemann Sums. This is not an analytical way to solve it; that is, you aren't going to have numbers left. You're going to solve it numerically, on a computer or calculator - you're going to plug in actual numbers.

Let's do an example. Let's say you're given the integral of *x*^2(sin(*x*))*dx*. Your first step would be to see if you can remember this integral or look it up in a table. Most likely, any integral you come up with is either in a table or it isn't solvable. There are whole books on writing out integrals like this, all the possible integrals you can think of that actually have solutions. But let's say you don't have that book handy, so the first thing you try is substitution: *u*=sin(*x*) and *du*=cos(*x*)*dx*. This doesn't make much sense, because if you plug in sin(*x*) you get *u* but *x*^2 becomes arcsin^2(*u*). You just made life a little more complicated, so maybe that's not the best way to do it. Because substitution is still the first thing you want to go to, what about using *u*=*x*^2 and *du*=2*xdx*. That's good, but you still have sin(*x*), which would become sin(square root of *u*) and that again sounds really complicated. So maybe substitution isn't the method you want to look at.

Since you have one function multiplied by another function, maybe you could do integration by parts. In integration by parts, I'm going to set this first function, *x*^2, equal to *u*. That leaves sin(*x*)*dx* as *dv*, because remember, the integral of *udv* = *uv* minus the integral of *vdu*. So my integral has to be *u* times *dv*: if *x*^2=*u*, then sin(*x*)*dx*=*dv*. If *u*=*x*^2, then *du*=2*xdx*. If *dv*=sin(*x*)*dx*, then *v*= -cos(*x*), because if I take the derivative of -cos(*x*), I end up with sin(*x*)*dx*. So I have *u*, *v*, *du* and *dv*. If I plug all of these into my equation for integration by parts, I get -*x*^2(cos(*x*)) + the integral of cos(*x*)2*xdx*. (The '+' comes from the minus sign of *v* combined with the minus sign in our equation.) I can rewrite this as -*x*^2(cos(*x*)) + 2 times the integral of *x*(cos(*x*))*dx*. I don't actually know the integral of *x*(cos(*x*))*dx* off the top of my head. It looks a little simpler than *x*^2(sin(*x*)), but it's still not something I just know.

Let's take a look at this whole equation: the integral of *x*^2(sin(*x*))*dx*= **- x^2(cos(x))** + 2 times the integral of

We just did a lot of work to find this, so let's make sure we did it correctly and take the derivative of the right-hand side. We have *x*^2(sin(*x*))*dx*= -*x*^2(cos(*x*)) + 2*x*(sin(*x*)) + 2cos(*x*) + *C*. Let's look at one term at a time. The derivative of -*x*^2(cos(*x*)), by the product rule, is the first times the derivative of the second plus the second times the derivative of the first: -*x*^2(-sin(*x*)) - 2*x*(cos(*x*). For my second term, 2*x*(sin(*x*), to find the derivative I'm again going to have to use the product rule. I get 2*x*(cos(*x*) + 2sin(*x*). It's getting longer and longer here! What about the third term, 2cos(*x*) + *C*? The derivative of that is just -2sin(*x*) + 0, because the derivative of a constant is zero.

Let's write this all out. I have *x*^2(sin(*x*)) - 2*x*(cos(*x*)) + 2*x*(cos(*x*)) + 2sin(*x*) - 2sin(*x*). This is fantastic. The 2*x*(cos(*x*)) terms cancel with one another, and the 2sin(*x*) terms cancel with one another. So the entire derivative simplifies down to *x*^2(sin(*x*)), which was my original integrand, so this indeed is the integral of *x*^2(sin(*x*))*dx*.

Let's review. At the end of the day, you have a few tools at your command to **solve problems of integration**. You can solve problems just by *memorization* of certain integrals. These are the really common integrals that you want to be able to pull off the top of your head, like sin(*x*), 1/*x* and *x*^2. You can also find integrals in a *table*, online or in a book.

But not all of these integrals can be found in one of these places, either in your head or some reference. So you need to use a couple of other tools, like for example *substitution*. Here you're taking an integral that depends on *x*, turning it into an integral that depends on *u* and hoping that new integral is easier to solve. You can also solve it by *parts*. This is like using the product rule in reverse. You can also use *algebra and trigonometry* to simplify your integrand, *f(x)*, into something that is more manageable. When you use trigonometry, you might want to use a different kind of substitution based on the rules of geometry and right triangles. Finally, if all of these fail, you can solve a definite integral *numerically*. That is, you can calculate the value of your integral using Riemann Sums.

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 Calculate Integrals of Exponential Functions 4:28
- 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
- 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

- Improving Customer Satisfaction & Retention
- Clinical Research: Help & Review
- SAT Literature: Help and Review
- LSAT Prep: Help and Review
- Common Core History & Social Studies Grades 9-10: Literacy Standards
- Holt United States History Chapter 9: A New National Identity (1812-1830)
- MTEL History: World War II
- Quiz & Worksheet - The Murder of Gonzago in Hamlet
- Quiz & Worksheet - Asthenosphere Properties
- Quiz & Worksheet - Nonmetal Elements on the Periodic Table
- Quiz & Worksheet - Inertial Frame of Reference
- Quiz & Worksheet - Verstehen in Sociology

- What is Dermal Tissue? - Definition & Function
- Gas Giants: Definition & Explanation
- Poetry Lesson Plan
- Scholarships for Study Abroad
- Fun & Easy Science Experiments for Kids
- When to Take the GMAT
- Phases of the Moon Lesson Plan
- Of Mice and Men Lesson Plan
- Mitosis Lesson Plan
- Descriptive Writing Prompts
- Trench Warfare Lesson Plan
- SAT Subject Test Registration Information

Browse by subject