Call Now to Set Up Tutoring:

(888) 888-0446

Calculus AB : Complete the Square with Integration

Study concepts, example questions & explanations for Calculus AB

Create An Account

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Complete The Square With Integration

What is complete the square?

Possible Answers:

Square the function

We draw the remaining sides of a square

Method for manipulating a quadratic equation into something able to be factored

Method for manipulating a quadratic equation for the quadratic equation

Correct answer:

Method for manipulating a quadratic equation into something able to be factored

Explanation:

Say we have the equation x^2 + bx = c . It is pretty hard to solve for in this way. Instead we can use a method called complete the square. Here we will add $\frac{b}{2}^2$ to both sides which will make it factorable.

x^2 + bx = c

$x^2 +bx +\frac{b}{2}^2 = c + \frac{b}{2}^2$

$(x+\frac{b}{2})^2 = c + \frac{b}{2}^2$

$x + \frac{b}{2} = \pm \sqrt{c+\frac{b}{2}^2}$

$x = -\frac{b}{2} \pm \sqrt{c+\frac{b}{2}^2}$

Report an Error

Example Question #2 : Complete The Square With Integration

Why would one need to use complete the square when integrating?

Possible Answers:

To find the roots of a function

When we need to manipulate an equation into a recognizable identity often for a difference of squares.

To use the quadratic formula

Only when we are asked to in a problem

Correct answer:

When we need to manipulate an equation into a recognizable identity often for a difference of squares.

Explanation:

While some of these answers can be obtained by using complete the square, they are not the goal of using complete the square to integrate. When we integrate, it is often hard to see when we may need to use a trigonometric identity or a log identity. Using complete the square, we are able to manipulate our integrals in order to see these identities.

Report an Error

Example Question #3 : Complete The Square With Integration

Use complete the square and show what the following equation would look like after it is applied x^2 -2x +5 = 0 .

Possible Answers:

x + 5 = 0

(x-2)^2 +4 = 0

(x-1)^2 +4 = 0

(x-1)^2 +5 = 0

Correct answer:

(x-1)^2 +4 = 0

Explanation:

To complete the square we can use one of two methods. We could use the formula x^2 + bx = c and add $\frac{b}{2}^2$ . Or if we recognize a factor that could work we can follow that route as well. So I am going to demonstrate the latter since the former is simply plugging into a formula. We will consider x^2 -2x + 5 = 0 . Now to factor our left hand side using complete the square we only consider x^2 - 2x .

x^2 - 2x looks similar to a common factorable equation x^2 - 2x + 1 . We can manipulate the equation by adding and subtracting in the same step.

x^2 - 2x + 1 - 1 = (x-1)^2 -1

We plug this back into our original equation:

x^2 - 2x +5 =0

(x-1)^2 - 1 +5 = 0

(x-1)^2 +4 = 0

Report an Error

Example Question #4 : Complete The Square With Integration

Show what the following equation will look like after completing the square. 0 = 1 - x^2 - 6x

Possible Answers:

0 = 10 - (x-3)^2

0 = 10 - (x+9)^2

0 = 1 - (x+3)^2

0 = 10 - (x+3)^2

Correct answer:

0 = 10 - (x+3)^2

Explanation:

In order to complete the square recognize that x^2 + 6x + 9 = (x + 3)^2 . We proceed as follows:

0 = 1 - x^2 - 6x

0 = 1 -(x^2 + 6x)

0 = 1 - (x^2 + 6x + 9 - 9)

0 = 10 - (x+3)^2

Report an Error

Example Question #5 : Complete The Square With Integration

Evaluate the following integral using complete the square.

$\int \frac{1}{x^2 + 10x +26}dx$

Possible Answers:

arcsec(x+5) + C

arctan(x+5) + C

arcsin(x+5) + C

arccos(x+5) + C

Correct answer:

arctan(x+5) + C

Explanation:

In order to complete the square for the quadratic we first consider the denominator of the integral function.

x^2 +10x + 26

x^2 + 10x + 25 - 25 + 26

(x + 5)^2 +1

Now our integral looks like $\int \frac{1}{(x+5)^2 +1} dx$ . We will use a u substitutions where u = x+5 . Now we have the integral:

$\int\frac{1}{u^2+1}dx$ .

We know that the integral $\int\frac{dx}{x^2 + 1} = arctan(x) + C$ . So our integral proceeds as follows:

$\int\frac{1}{u^2 + 1}dx = arctan(u) + C = arctan(x+5) + C$ .

Report an Error

Example Question #6 : Complete The Square With Integration

What would be the proper difference of squares to use to integrate the following to be equal to $\int\frac{dx}{1 - x^2 - 4x} = arcsin(\frac{x+2}{\sqrt{5}}) + C$ ?

Possible Answers:

5^2 - (x+2)^2

$\sqrt(5)^2 - (x+2)^2$

4 - (x+2)^2

1 - (x+2)^2

Correct answer:

$\sqrt(5)^2 - (x+2)^2$

Explanation:

If we want to integrate the following to equal $arcsin(\frac{x+2}{\sqrt{5}}) +C$ we first must recall that $\int\frac{dx}{a^2 - x^2} = arcsin(\frac{x}{a}) + c$ . We must consider the denominator the integral:

1 - x^2 - 4x

1 - (x^2 +4x)

1 -(x^2 + 4x +4 - 4)

5 -(x^2 + 4x + 4)

5 -(x + 2)^2

Now we use a u substitution where u = x+2

5 - u^2

We also know that $\sqrt(5)^2 = 5$ so we can substitute that in as well

$\sqrt(5)^2 - u^2$

If we plug this back into our integral we can now solve this integral easily.

$\int\frac{dx}{\sqrt(5)^2 - u^2} = arcsin(u) + C = arcsin(\frac{x+2}{\sqrt{5}}) + C$ . The difference of two squares that we needed was $\sqrt(5) - (x+2)^2$

Report an Error

Example Question #7 : Complete The Square With Integration

Use complete the square to solve the following integral

$\int\frac{dx}{x^2+6x+7}$

Possible Answers:

$ln|(x+3) + \sqrt{(x+3)^2 - \sqrt{2}} + C$

$ln|(x+3) + \sqrt{(x+3)^2 - 2} + C$

$arcsin(\frac{(x+3)^2 }{\sqrt{2}}) + C$

$arctan(\frac{(x+3)^2 }{\sqrt{2}}) + C$

Correct answer:

$ln|(x+3) + \sqrt{(x+3)^2 - 2} + C$

Explanation:

To solve this integral, we must first complete the square of the denominator.

x^2 + 6x + 7

x^2 + 6x + 9 - 9 +7

(x+3)^2 - 2

Plugging this back into our integral we have $\int\frac{dx}{(x+3)^2 - 2}$ . Recall that $\int\frac{dx}{x^2 \pm a^2} = ln|x + \sqrt{x^2 \pm a^2}|$

$\int\frac{dx}{(x+3)^2 - 2}$

$\int\frac{dx}{(x+3)^2 - \sqrt{2}^2}$

Using a u substitution we will let u = x+2

$\int\frac{dx}{u^2 - \sqrt{2}^2} = ln|u + \sqrt{u^2 - \sqrt{2}^2}| + C = ln|(x+3) + \sqrt{(x+3)^2 - 2}| + C$

Report an Error

Example Question #8 : Complete The Square With Integration

Use complete the square to solve the following integral

$\int\frac{dx}{-3 -x^2 -4x}$

Possible Answers:

$ln|\frac{3 + x}{x-1}| + C$

$\frac{1}{2}ln|\frac{3 + x}{x-1}| + C$

$\frac{1}{2}ln|\frac{3 - x}{x+1}| + C$

$\frac{1}{2}ln|\frac{x-1}{3+x}| + C$

Correct answer:

$\frac{1}{2}ln|\frac{3 + x}{x-1}| + C$

Explanation:

We start by considering the denominator.

-3 - x^2-4x

-3 - (x^2 +4x)

-3 - (x^2 +4x +4 -4)

1 - (x^2 +4x +4)

1 - (x+2)^2

Plugging this back into our integral we get $\int\frac{dx}{1-(x+2)^2$ . Using a u substitution, we will let u = x+2 .

$\int\frac{dx}{1 - u^2}$

Recall that $\int\frac{dx}{a^2 - x^2} = \frac{1}{2a}ln|\frac{a+x}{a-x}|+C$

$\int\frac{dx}{1 - u^2}$ $= \frac{1}{2}ln|\frac{1+u}{1-u}| +C = \frac{1}{2}ln|\frac{3 + x}{x-1} + C$

Report an Error

Example Question #9 : Complete The Square With Integration

True or False: We can only use complete the square if it gives us an exact formula for a solution to the integral.

Possible Answers:

True

False

Correct answer:

False

Explanation:

This is not true. We use complete the square in order to get the integral into a familiar form but often times we need to use a u substitution in order to be able to complete the problem.

Report an Error

Example Question #10 : Complete The Square With Integration

True or False: We are able to complete the square by adding and subtracting the same term to our given equation. This is basically adding zero, and therefore it does not change the value of our equation.

Possible Answers:

False

True

Correct answer:

True

Explanation:

Take x^2 -2x +5 = 0 for example. We are able to complete the square by adding and subtracting . When we do this, it does not change the value of our equation and this is because of and are identities of each other meaning that 1 + (-1) = 0 . By doing this with any number (adding and subtracting the same number) we are able to successfully complete the square without changing the value of our equation.

Report an Error

View Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

View Tutors

Ashna
Certified Tutor

guru nanak dev university , Master's/Graduate, Commerce .

View Tutors

Paul J
Certified Tutor

The University of Toledo, Bachelor of Science, Multi-/Interdisciplinary Studies, General. Eastern Illinois University, Master...

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

English Tutors in Phoenix, Computer Science Tutors in San Francisco-Bay Area, Reading Tutors in Dallas Fort Worth, GRE Tutors in New York City, English Tutors in Los Angeles, MCAT Tutors in Los Angeles, Computer Science Tutors in Phoenix, Statistics Tutors in Denver, English Tutors in Miami, SSAT Tutors in Atlanta

Popular Courses & Classes

MCAT Courses & Classes in Philadelphia, SAT Courses & Classes in San Diego, Spanish Courses & Classes in Philadelphia, MCAT Courses & Classes in Seattle, GMAT Courses & Classes in Atlanta, SAT Courses & Classes in Los Angeles, MCAT Courses & Classes in Washington DC, ACT Courses & Classes in Seattle, SSAT Courses & Classes in Dallas Fort Worth, ISEE Courses & Classes in Boston

Popular Test Prep

ISEE Test Prep in Los Angeles, SSAT Test Prep in Los Angeles, MCAT Test Prep in Seattle, MCAT Test Prep in Houston, SAT Test Prep in Boston, MCAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Seattle, SAT Test Prep in Los Angeles, ACT Test Prep in Seattle, GMAT Test Prep in Washington DC

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus AB : Complete the Square with Integration

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Complete The Square With Integration

Example Question #2 : Complete The Square With Integration

Example Question #3 : Complete The Square With Integration

Example Question #4 : Complete The Square With Integration

Example Question #5 : Complete The Square With Integration

Example Question #6 : Complete The Square With Integration

Example Question #7 : Complete The Square With Integration

Example Question #8 : Complete The Square With Integration

Example Question #9 : Complete The Square With Integration

Example Question #10 : Complete The Square With Integration

All Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: