Back To Course

AP Calculus AB & BC: Homework Help Resource17 chapters | 139 lessons

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 55,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you'll learn about the different types of integration problems you may encounter. You'll see how to solve each type and learn about the rules of integration that will help you.

Integrating various types of functions is not difficult. All you need to know are the rules that apply and how different functions integrate. You know the problem is an integration problem when you see the following symbol:

Remember, too, that your integration answer will always have a constant of integration, which means that you are going to add '+ *C*' for all your answers. The various types of functions you will most commonly see are monomials, reciprocals, exponentials, and trigonometric functions. Certain rules like the constant rule and the power rule will also help you. Let's start with monomials.

**Monomials** are functions that have only one term. Some monomials are just constants, while others also involve variables. None of the variables have powers that are fractions; all the powers are whole integers. For example, *f*(*x*) = 6 is a constant monomial, while *f*(*x*) = *x* is a monomial with a variable.

When you see a constant monomial as your function, the answer when you integrate is our constant multiplied by the variable, plus our constant of integration. For example, if our function is *f*(*x*) = 6, then our answer will be the following:

We can write this in formula form as the following:

If our function is a monomial with variables like *f*(*x*) = *x*, then we will need the aid of the power rule which tells us the following:

The **power rule** tells us that if our function is a monomial involving variables, then our answer will be the variable raised to the current power plus 1, divided by our current power plus 1, plus our constant of integration. This is only if our current power is not -1. For example, if our function is *f*(*x*) = *x*, where our current power is 1, then our answer will be this:

Recall that if you don't see a power, it is always 1 because anything raised to the first power is itself.

Let's try another example. If our function is *f*(*x*) = *x*^2, then our answer will be the following:

Whatever our current power is our answer will be the variable raised to the next power divided by the next power. In the above example, our current power is 2, so our next power is 3. In our answer, we have a 3 for the variable's power and for the denominator following the power rule. If our monomial is a combination of a constant and a variable, we have the constant rule to help us. The constant rule looks like this:

The **constant rule** tells us to move the constant out of the integral and then to integrate the rest of the function. For example, if our function is *f*(*x*) = 6*x*, then our integral and answer will be the following:

We've moved the 6 outside of the integral according to the constant rule, and then we integrated the *x* by itself using the power rule. For the answer, we simplified the 6*x*^2/2 to 3*x*^2 since 6*x* divides evenly by 2.

Another type of function we will deal with is the reciprocal. The integral of the reciprocal follows this formula:

The formula is telling us that when we integrate the reciprocal, the answer is the natural log of the absolute value of our variable plus our constant of integration. Exponential functions include the *e*^*x* function as well as the log(*x*) function and these types of functions follow these formulas for integration:

The first formula tells us that when we have a function *e*^*x*, our answer for the integral will be *e*^*x* + *C*. The *a* in the middle integral formula stands for a constant. The middle formula tells us that when we have, for example, a function like 3^*x*, then our answer after integrating will be 3^*x*/log(3) + *C*. The integral formula tells us that the integral of the natural log of *x* function is *x*(log(*x*) - 1) plus our constant of integration.

Our trigonometric functions include cosine, sine, and secant functions. They follow these formulas:

If you are integrating the cosine function, you will end up with the sine function plus the constant of integration. Integrating the sine function gives you the negative cosine function plus our constant of integration. If you see the secant function squared, your answer will be the tangent function plus our constant of integration.

Let's review. Integrating different functions involves referring to the formulas for each type of function along with applying the constant or power rule when necessary. The **constant rule** tells us to move the constant out of the integral and then to integrate the rest of the function. The **power rule** tells us that if our function is a monomial involving variables, then our answer will be the variable raised to the current power plus 1, divided by our current power plus 1, plus our constant of integration. Always remember your constant of integration when integrating.

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 95 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.

You are viewing lesson
Lesson
13 in chapter 13 of the course:

Back To Course

AP Calculus AB & BC: Homework Help Resource17 chapters | 139 lessons

- 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
- How to Solve Improper Integrals 11:01
- Integration Problems in Calculus: Solutions & Examples 5:14
- Go to Integration and Integration Techniques in AP Calculus: Homework Help Resource

- Pharmacology & Pain Treatment
- Pharmacology & Mental Health
- Pharmacology & Cardiovascular Health
- PLACE Special Education Generalist: Practice & Study Guide
- MTTC Physical & Other Health Impairments: Practice & Study Guide
- Effects of Impairments on Development & Learning
- IEP Teams & Meetings
- Basics of the Individualized Instruction Plan (IEP)
- Medications for Colds & Coughs
- Alternative Medicine for Pain Treatment
- Teacher Retirement System of Texas Withdrawal
- New York State Science Standards
- Third Grade New York State Science Standards
- Wisconsin State Teaching Standards
- New York State Science Standards for Grade 4
- Arizona English Language Proficiency Standards
- Arizona English Language Proficiency Standards & Levels

- How Call Center Technology Increases Customer Satisfaction
- Measuring Emotional Intelligence: Tools & Benefits
- Customer Lifetime Value (CLV): Definition, Calculation & Uses
- How to Delight Customers: Principle & Examples
- Short Stories about Heaven for Kids
- Secure, Safe & Orderly Schools: Definition & Characteristics
- The Armadillo from Amarillo Lesson Plan
- Bromothymol Blue: Preparation & Safety
- Quiz & Worksheet - Emotional Intelligence & Business Success
- Quiz & Worksheet - Real Estate Specialization
- Quiz & Worksheet - Transferring Knowledge at Work
- Quiz & Worksheet - Legal & Ethical Concerns in Decision Making
- Quiz & Worksheet - Pitfalls of Employee Recognition Programs
- Hypothesis Testing in Statistics Flashcards
- Summarizing Data Flashcards

- 8th Grade World History: Enrichment Program
- History 104: US History II
- AP English Literature: Homeschool Curriculum
- 12th Grade English Textbook
- Supplemental English: Study Aid
- PSAT Math - Functions: Help and Review
- High School Algebra: Data, Statistics, and Probability
- Quiz & Worksheet - Aggregate Fruit
- Quiz & Worksheet - Characteristics of Transnational Corporations
- Quiz & Worksheet - Settlement & Conflict at Jamestown
- Quiz & Worksheet - RNA Primer in DNA Replication
- Quiz & Worksheet - History of the Cult of Mithras

- Utopia by Thomas More: Summary & Analysis
- What is the Plot of a Story? - Definition & Examples
- Good Persuasive Writing Topics for Middle School
- FAFSA Work Study Deadline
- Russian Revolution Lesson Plan
- Feudalism Lesson Plan
- First Grade Writing Prompts
- 7th Grade Writing Prompts
- Fun Writing Prompts
- In-School Resources for Teacher Professional Development
- Independent Study Ideas
- States of Matter Lesson Plan

Browse by subject