Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Antiderivatives by substitution of variables

Study concepts, example questions & explanations for AP Calculus AB

Create An Account

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Antiderivatives By Substitution Of Variables

Use a change of variable (aka a u-substitution) to evaluate the integral,

$\int x\sqrt{1+x^2}dx$

Possible Answers:

$\frac{1}{3}x^{\frac{3}{2}}+C$

$\frac{1}{6}(2x+1)^{\frac{2}{3}}+C$

$\frac{3}{4}(x^2+1)^{\frac{3}{2}}+C$

$\frac{1}{3}(x^2+1)^{\frac{3}{2}}+C$

$\frac{1}{6}(x^2+1)^{\frac{2}{3}}+\sqrt{x}+C$

Correct answer:

$\frac{1}{3}(x^2+1)^{\frac{3}{2}}+C$

Explanation:

$\int x\sqrt{1+x^2}dx$

Integrals such as this are seen very commonly in introductory calculus courses. It is often useful to look for patterns such as the fact that the polynomial under the radical in our example, 1+x^2 , happens to be one order higher than the factor outside the radical, You know that if you take a derivative of a second order polynomial you will get a first order polynomial, so let's define the variable:

u = 1+x^2 (1)

Now differentiate with respect to to write the differential for ,

du = 2x dx (2)

Looking at equation (2), we can solve for xdx , to obtain $xdx =\frac{1}{2}du$ . Now if we look at the original integral we can rewrite in terms of

$\int x\sqrt{1+x^2}dx = \int\sqrt{1+x^2}\underbrace{xdx} = \int\sqrt{u}\left(\frac{1}{2}du \right )$

Now proceed with the integration with respect to .

$\int\sqrt{u}\left(\frac{1}{2}du \right ) = \frac{1}{2}\int u^{\frac{1}{2}}du$

$= \frac{1}{2}\left[\frac{1}{\frac{1}{2}+1}u^{\frac{1}{2}+1} \right ]+C$

$=\frac{1}{2} \left(\frac{2}{3}u^{\frac{3}{2}} \right )+C$

$=\frac{1}{3}u^{\frac{3}{2}}+C$

Now write the result in terms of using equation (1), we conclude,

$\int x\sqrt{1+x^2}dx=\frac{1}{3}(x^2+1)^{\frac{3}{2}}+C$

Report an Error

Example Question #2 : Antiderivatives By Substitution Of Variables

Use u-subtitution to fine

$\int \sin^2(x)\cos (x)dx$

Possible Answers:

$\frac{1}{6}sin^3(x)cos^2(x)$

$\frac{1}{4}sin^4(x)$

$\frac{-1}{6}cos^3(x)sin^2(x)$

$\frac{1}{6}sin^2(x)$

$\frac{1}{3}sin^3(x)$

Correct answer:

$\frac{1}{3}sin^3(x)$

Explanation:

Let u=sin(x)

Then du=cos(x)dx

Now we can subtitute

$\int sin^2(x) \cos(x) dx=\int u^2du= \frac{1}{3}u^3$

Now we substitute back

$\frac{1}{3}u^3=\frac{1}{3} \sin^3(x)$

Report an Error

Example Question #3 : Antiderivatives By Substitution Of Variables

Evaluate $\int_0^1 \frac{6x}{1+x^2} dx$

Possible Answers:

$3\ln2$

$\ln3$

$2\ln3$

$\ln2$

Correct answer:

$3\ln2$

Explanation:

We can use substitution for this integral.

Let u = 1+x^2 ,

then du = 2x dx .

Multiplying this last equation by , we get 3du = 6x dx .

Now we can make our substitutions

$\int_0^1 \frac{6x}{1+x^2} dx$ . Start

$=\int_1^2 \frac{3}{u}du$ . Swap out 1+x^2 with , and 6x dx with 3du . Make sure you also plug the bounds on the integral into u = 1+x^2 for to get the new bounds.

$=3\int_1^2 \frac{1}{u}du$ . Factor out the .

$=3 \ln(u)|_1^2$ . Integrate (absolute value signs are not needed since $1\le u \le2$ .)

$=3(\ln(2)-\ln(1))$ . Evaluate

$=3\ln2 -0 = 3\ln2$ .

Report an Error

Example Question #4 : Antiderivatives By Substitution Of Variables

Solve the following integral using substitution:

$\int{x^3(2+x^4)^5}dx$

Possible Answers:

$\frac{1}{6}(2+x^4)^6+c$

$\frac{1}{2}x^4+\frac{1}{24}x^{24}+c$

$2x+x^{20}+c$

$2x^3+x^{23}+c$

$\frac{1}{24}(2+x^4)^6+c$

Correct answer:

$\frac{1}{24}(2+x^4)^6+c$

Explanation:

To solve the integral, we have to simplify it by using a variable u to substitute for a variable of x.

For this problem, we will let u replace the expression 2+x^4 .

Next, we must take the derivative of u. Its derivative is du=4x^3dx .

Next, solve this equation for dx so that we may replace it in the integral.

Plug in place of (2+x^4) and $\frac{du}{4x^3}$ in place of into the original integral and simplify.

The x^3 in the denominator cancels out the remaining x^3 in the integral, leaving behind a $\frac{1}{4}$ . We can pull the $\frac{1}{4}$ out front of the integral. Next, take the anti-derivative of the integrand and replace u with the original expression, adding the constant to the answer.

The specific steps are as follows:

1. $\int{x^3(2+x^4)^5dx}$

2. u=2+x^4

3. du=4x^3dx

4. $dx=\frac{du}{4x^3}$

5. $\int{x^3u^5(\frac{du}{4x^3})}$

6. $\frac{1}{4}\int{u^5}du$

7. $\frac{1}{4}(\frac{1}{6}u^6)$

8. $\frac{1}{4}(\frac{1}{6}(2+x^4)^6$

9. $\frac{1}{24}(2+x^4)^6+c$

Report an Error

Example Question #5 : Antiderivatives By Substitution Of Variables

Solve the following integral using substitution:

$\int{x^2\sqrt{x^3+1}}dx$

Possible Answers:

$\frac{2}{3}(x^3+1)^9+c$

$x^{\frac{7}{2}}+x^2+c$

$\frac{1}{3}(x^3+1)^{\frac{5}{2}}+c$

$\frac{2}{9}(x^3+1)^{\frac{3}{2}}+c$

$\sqrt[3]{(x^3+1)}(\frac{1}{x^2})+c$

Correct answer:

$\frac{2}{9}(x^3+1)^{\frac{3}{2}}+c$

Explanation:

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression x^3+1 . Next, we must take the derivative of u. Its derivative is 3x^2dx . Next, solve this equation for so that we may replace it in the integral. Plug in place of x^3+1 and $\frac{du}{3x^2}$ in place of into the original integral and simplify. The x^2 in the denominator cancels out the remaining x^2 in the integral, leaving behind a $\frac{1}{3}$ . We can pull the $\frac{1}{3}$ out front of the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{x^2\sqrt{x^3+1}}dx$

2. = x^3+1

3. du=3x^2dx

4. $\int{x^2\sqrt{u}\frac{du}{3x^2}}$

5. $\frac{1}{3}\int{\sqrt{u}du}$

6. $\frac{1}{3}(\frac{2}{3}u^{\frac{3}{2}})+c$

7. $\frac{2}{9}u^{\frac{3}{2}}+c$

8. $\frac{2}{9}(x^3+1)^{\frac{3}{2}}+c$

Report an Error

Example Question #6 : Antiderivatives By Substitution Of Variables

Solve the following integral using substitution:

$\int{cos^3xsinxdx}$

Possible Answers:

$\frac{-cos^4x}{4}+c$

$\frac{-1}{3}(cosx)^3+c$

$\frac{sin^4x}{4}+c$

cotxcos^2x+c

$\frac{1}{3}(cosx)^3+c$

Correct answer:

$\frac{-cos^4x}{4}+c$

Explanation:

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression cosx . Next, we must take the derivative of . Its derivative is -sinxdx . Next, solve this equation for so that we may replace it in the integral. Plug in place of cosx and $\frac{du}{-sinx}$ in place of into the original integral and simplify. The sinx in the denominator cancels out the remaining sinx in the integral, leaving behind a . Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{cos^3xsinxdx}$

2. u=cosx

3. du=-sinxdx

4. $\int{u^3sinx(\frac{du}{-sinx})}$

5. $\int-u^3du$

6. $\frac{-1}{4}u^4+c$

7. $\frac{-cos^4x}{4}+c$

Report an Error

Example Question #7 : Antiderivatives By Substitution Of Variables

Solve the following integral using substitution:

$\int{xsin(x^2)dx}$

Possible Answers:

2cos(x^2)+c

$\frac{1}{2}sin(x^2)+c$

$\frac{-1}{2}sin(x^2)+c$

$\frac{1}{2}cos(x^2)+c$

$\frac{-1}{2}cos(x^2)+c$

Correct answer:

$\frac{-1}{2}cos(x^2)+c$

Explanation:

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression x^2 . Next, we must take the derivative of . Its derivative is . Next, solve this equation for so that we may replace it in the integral. Plug in place of x^2 and $\frac{du}{2x}$ in place of into the original integral and simplify. The in the denominator cancels out the remaining in the integral, leaving behind a $\frac{1}{2}$ . We can pull the $\frac{1}{2}$ out front of the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{xsin(x^2)dx}$

2. u=x^2

3. du=2xdx

4. $\int{x(sinu)\frac{du}{2x}}$

5. $\frac{1}{2}\int{sinudu}$

6. $\frac{-1}{2}cosu+c$

7. $\frac{-1}{2}cos(x^2)+c$

Report an Error

Example Question #2 : Antiderivatives By Substitution Of Variables

Solve the following integral using substitution:

$\int{x^2e^{x^3}dx}$

Possible Answers:

$\frac{1}{2}e^{x2}+c$

$\frac{3}{2}e^{x^2}+c$

$\frac{1}{3}e^{x^3}+c$

$\frac{3}{2}e^{x^3}+c$

$2xe^{x^3}+c$

Correct answer:

$\frac{1}{3}e^{x^3}+c$

Explanation:

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression x^3 . Next, we must take the derivative of . Its derivative is 3x^2 . Next, solve this equation for so that we may replace it in the integral. Plug in place of x^3 and $\frac{du}{3x^2}$ in place of into the original integral and simplify. The x^2 in the denominator cancels out the remaining x^2 in the integral, leaving behind a $\frac{1}{3}$ . We can pull the $\frac{1}{3}$ out front of the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{x^2e^{x^3}dx}$

2. u=x^3

3. du=3x^2

4. $\int{x^2e^u(\frac{du}{3x^2})}$

5. $\frac{1}{3}\int{e^udu}$

6. $\frac{1}{3}e^u+c$

7. $\frac{1}{3}e^{x^3}+c$

Report an Error

Example Question #9 : Antiderivatives By Substitution Of Variables

Solve the following integral using substitution:

$\int{e^xcos(e^x)dx}$

Possible Answers:

-cos(e^x)+c

-sin(e^x)+c

sin(e^x)+c

$e^{cosx}+c$

$e^{sinx}+c$

Correct answer:

sin(e^x)+c

Explanation:

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression e^x . Next, we must take the derivative of . Its derivative is e^x . Next, solve this equation for so that we may replace it in the integral. Plug in place of e^x and $\frac{du}{e^x}$ in place of into the original integral and simplify. The e^x in the denominator cancels out the remaining e^x in the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{e^xcos(e^x)dx}$

2. u=e^x

3. du=e^xdx

4. $\int{e^xcosu(\frac{du}{e^x})}$

5. $\int{cosudu}$

6. sinu+c

7. sin(e^x)+c

Report an Error

Example Question #9 : Antiderivatives By Substitution Of Variables

Solve the following integral using substitution:

$\int{\frac{sin(lnx)}{x}dx}$

Possible Answers:

$sinx(\frac{1}{x})+c$

-cos(lnx)+c

cos(lnx)+c

$\frac{1}{\sqrt{1-lnx}}+c$

$-sinx(\frac{1}{x})+c$

Correct answer:

-cos(lnx)+c

Explanation:

To solve the integral, we have to simplify it by using a variable to substitute for a variable of . For this problem, we will let u replace the expression lnx . Next, we must take the derivative of . Its derivative is $\frac{1}{x}$ . Next, solve this equation for so that we may replace it in the integral. Plug in place of lnx and dux in place of into the original integral and simplify. The in the denominator cancels out the remaining in the integral. Next, take the anti-derivative of the integrand and replace with the original expression, adding the constant to the answer. The specific steps are as follows:

1. $\int{\frac{sin(lnx)}{x}dx}$

2. u=lnx

3. $du=(\frac{1}{x}dx)$

4. dux=dx

5. $\int{\frac{sinu}{x}dux}$

6. $\int{sinudu}$

7. -cosu+c

8. -cos(lnx)+c

Report an Error

View AP Calculus AB Tutors

Anne
Certified Tutor

University of Maryland-Baltimore, Doctor of Science, Accounting.

View AP Calculus AB Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

View AP Calculus AB Tutors

Pankaj
Certified Tutor

University of Delhi, Electrical Engineer, Electrical Engineering. Columbia University in the City of New York, Master of Arts...

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

LSAT Tutors in Dallas Fort Worth, SAT Tutors in New York City, Algebra Tutors in San Diego, Reading Tutors in San Francisco-Bay Area, Reading Tutors in Phoenix, Computer Science Tutors in Philadelphia, LSAT Tutors in Washington DC, Calculus Tutors in Philadelphia, Spanish Tutors in Phoenix, French Tutors in Seattle

Popular Courses & Classes

GMAT Courses & Classes in Miami, GMAT Courses & Classes in Chicago, MCAT Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Miami, ISEE Courses & Classes in Phoenix, ISEE Courses & Classes in Miami, ISEE Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in San Diego, LSAT Courses & Classes in Houston, ACT Courses & Classes in Los Angeles

Popular Test Prep

SSAT Test Prep in Phoenix, GRE Test Prep in Seattle, GRE Test Prep in San Diego, MCAT Test Prep in Atlanta, MCAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Miami, GMAT Test Prep in Chicago, MCAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Miami, LSAT 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

AP Calculus AB : Antiderivatives by substitution of variables

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1 : Antiderivatives By Substitution Of Variables

Example Question #2 : Antiderivatives By Substitution Of Variables

Example Question #3 : Antiderivatives By Substitution Of Variables

Example Question #4 : Antiderivatives By Substitution Of Variables

Example Question #5 : Antiderivatives By Substitution Of Variables

Example Question #6 : Antiderivatives By Substitution Of Variables

Example Question #7 : Antiderivatives By Substitution Of Variables

Example Question #2 : Antiderivatives By Substitution Of Variables

Example Question #9 : Antiderivatives By Substitution Of Variables

Example Question #9 : Antiderivatives By Substitution Of Variables

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: