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 #11 : Antiderivatives By Substitution Of Variables

Solve the following integral using substitution:

$\int{(x^2+e^{3x})(x^3+e^{3x})^{\frac{6}{7}}dx}$

Possible Answers:

$\frac{7}{39}(x^3+e^{3x})^{\frac{13}{7}}+c$

$\frac{12}{7}(x^3+e^{3x})^{\frac{7}{12}}+c$

$-21(x^3+e^{3x})^{\frac{1}{3}}+c$

$\frac{39}{7}(x^3+e^{3x})^{\frac{7}{6}}+c$

$3x^2(e^{3x})^{\frac{-1}{7}}+c$

Correct answer:

$\frac{7}{39}(x^3+e^{3x})^{\frac{13}{7}}+c$

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 $x^3+e^{3x}$ . Next, we must take the derivative of . Its derivative is $3x^2+e^{3x}$ . Next, solve this equation for so that we may replace it in the integral. Plug in place of $x^3+e^{3x}$ and $\frac{du}{3x^2+e^{3x}}$ in place of into the original integral and simplify. The $x^2+e^{3x}$ in the denominator cancels out the remaining $x^2+e^{3x}$ 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. $\int{(x^2+e^{3x})(x^3+e^{3x})^{\frac{6}{7}}dx}$

2. $u=x^3+e^{3x}$

3. $du=(3x^2+3e^{3x})dx$

4. $\int(x^2+e^{3x})u^{\frac{6}{7}(\frac{du}{x^2+e^{3x}})}$

5. $\frac{1}{3}\int{u^{\frac{6}{7}}}du$

6. $\frac{1}{3}(\frac{7}{13})u^{\frac{13}{7}}$

7. $\frac{7}{39}u^{\frac{13}{7}}+c$

8. $\frac{7}{39}(x^3+e^{3x})^{\frac{13}{7}}+c$

Report an Error

Example Question #12 : Antiderivatives By Substitution Of Variables

$\int e^{4x-2}dx=?$

Possible Answers:

$4e^{4x-2}+C$

$4e^{4x-2}$

$\frac{1}{4}e^{4x-2}+C$

$e^{4x-2}+C$

$\frac{1}{4}e^{4x-2}$

Correct answer:

$\frac{1}{4}e^{4x-2}+C$

Explanation:

This is a u-substitution integral. We need to substitute the new function, which is modifying our base function (the exponential).

$\int e^xdx=e^x+C$ , but instead of that, our problem is e^u . We can solve this integral by completing the substitution.

$u=4x-2.\; du=4dx. \;dx=\frac{du}{4}.$

Now, we can replace everything in our integrand and rewrite in terms of our new variables:

$\int e^{4x-2}dx=\int e^u\frac{du}{4}=\frac{1}{4}\int e^udu.$

$=\frac{1}{4}e^u+C=\frac{1}{4}e^{4x-2}+C$ .

Remember to plug your variable back in and include the integration constant since we have an indefinite integral.

Report an Error

Example Question #13 : Antiderivatives By Substitution Of Variables

$\int 2cos(2x)dx=?$

Possible Answers:

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

4sin(2x)+C

sin(2x)+C

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

-sin(2x)+C

Correct answer:

sin(2x)+C

Explanation:

This is a u-substitution integral. We need to choose the following substitutions:

$u=2x.\; du=2dx.$

Now, we can replace our original problem with our new variables:

$\int2cos(2x)dx=\int cos(2x)2dx=\int cos(u)du$

=sin(u)+C=sin(2x)+C.

In the last step, we need to plug in our original function and add the integration constant.

Report an Error

Example Question #14 : Antiderivatives By Substitution Of Variables

$\int \sqrt{3+4x}dx=?$

Possible Answers:

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

$\frac{1}{6}(3+4x)^{3/2}+C$

$(3+4x)^{3/2}+C$

$\frac{3}{8}(3+4x)^{3/2}+C$

$\frac{8}{3}(3+4x)^{3/2}+C$

Correct answer:

$\frac{1}{6}(3+4x)^{3/2}+C$

Explanation:

This is a hidden u-substitution problem! Because we have a function under our square root, we cannot just simply integrate it. Therefore, we need to choose the function under the square root as our substitution variable!

$u=3+4x.\; du=4dx.\; dx=\frac{du}{4}.$

Now, let us rewrite our original equation in terms of our new variable!

$\int \sqrt{3+4x}dx=\int \sqrt{u}\frac{du}{4}=\frac{1}{4}\int \sqrt udu.$

$=\frac{1}{4}\int u^{1/2}=\frac{1}{4}\frac{u^{3/2}}{3/2}+C=\frac{1}{4}*\frac{2}{3}u^{3/2}+C$

$=\frac{2}{12}u^{3/2}+C=\frac{1}{6}({3+4x)}^{3/2}+C$ .

Report an Error

Example Question #15 : Antiderivatives By Substitution Of Variables

$\int (2x-1)(x^2-x+8)^7dx=?$

Possible Answers:

${(x^2-x+8)^8}+C$

$\frac{(x^2-x+8)^8}{8}+C$

$\frac{(\frac{1}{3}x^3-\frac{x^2}{2}+8x)^8}{8}+C$

8(x^2-x+8)^8+C

Correct answer:

$\frac{(x^2-x+8)^8}{8}+C$

Explanation:

This is a u-substitution problem. We need to find a function and its derivative in the integral.

$u=x^2-x+8.\; du=(2x-1)dx.$

Now, replace your variables, and integrate.

$\int (2x-1)(x^2-x+8)^7dx=\int (x^2-x+8)^7(2x-1)dx$

$=\int u^7du=\frac{u^8}{8}+C=\frac{(x^2-x+8)^8}{8}+C$ .

Report an Error

Example Question #16 : Antiderivatives By Substitution Of Variables

$\int x\sqrt{3+x}dx=?$

Possible Answers:

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

$\frac{x^3}{3}+\frac{x^4}{4}+C$

$\frac{2}{5}(3+x)^{5/2}+C$

$\frac{2}{5}(3+x)^{5/2}-2(3+x)^{3/2}+C$

$\sqrt{3+x}+\frac{x}{2\sqrt{3+x}}+C$

Correct answer:

$\frac{2}{5}(3+x)^{5/2}-2(3+x)^{3/2}+C$

Explanation:

This problem is an application of the u-substitution method.

$u=3+x.\; x=u-3\; du=1dx.$

Now, be careful that you replace everything in the original integral in terms of our new variables. This includes the term!

$\int x\sqrt{3+x}dx=\int(u-3)u^{1/2}du=\int u^{3/2}-3u^{1/2}du$

$=\frac{u^{5/2}}{5/2}-3\frac{u^{3/2}}{3/2}=\frac{2}{5}u^{5/2}-2u^{3/2}+C$

$=\frac{2}{5}(3+x)^{5/2}-2(3+x)^{3/2}+C$ .

Report an Error

Example Question #951 : Ap Calculus Ab

$\int \frac{2x-4}{2x^2-8x+3}dx=?$

Possible Answers:

2ln|4x-8|+C

$\frac{1}{2}ln|4x-8|+C$

$\frac{1}{2}ln|2x^2-8x+3|+C$

ln|2x^2-8x+3|+C

2ln|2x^2-8x+3|+C

Correct answer:

$\frac{1}{2}ln|2x^2-8x+3|+C$

Explanation:

To simplify the integral, we need to substitute new variables:

$\\u=2x^2-8x+3.\; \\du=(4x-8)dx \\du=2(2x-4)dx \\\frac{1}{2}du=(2x-4)dx$

Now, we can replace our original variables, and integrate!

$\int \frac{2x-4}{2x^2-8x+3}dx=\int \frac{du}{2u}=\frac{1}{2}\int\frac{du}{u}$

$\\=\frac{1}{2}ln|u|+C \\\\ =\frac{1}{2}ln{|2x^2-8x+3|}+C$ .

Report an Error

Example Question #952 : Ap Calculus Ab

$\int \frac{2lnx}{x}dx=?$

Possible Answers:

(lnx)^2+C

2lnx^2+C

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

2ln(x)+C

$\frac{2}{lnx}+C$

Correct answer:

(lnx)^2+C

Explanation:

This is a hidden u-substitution problem! Remember, to use substitution, we need to have an integral where a function and its derivative live inside. If you look closely, you will see we have just that!

$u=lnx.\; du=\frac{1}{x}dx.$

Now, rewrite the integral, and integrate:

$\int \frac{2lnx}{x}==2\int lnx*\frac{1}{x}dx=2\int u du=2\frac{u^2}{2}+C$

=u^2+C=(lnx)^2+C.

Report an Error

Example Question #61 : Techniques Of Antidifferentiation

Integrate:

$\int 5y^2\cos(y^3)dy$

Possible Answers:

$\frac{5}{3}\sin(y^3)$

$\frac{5y^3}{3}\sin(y^3)+C$

$\frac{1}{15}\sin(y^3)+C$

$\frac{5}{3}\sin(y^3)+C$

$-\frac{5}{3}\sin(y^3)+C$

Correct answer:

$\frac{5}{3}\sin(y^3)+C$

Explanation:

To integrate, we must make the following substitution:

u=y^3, du=3y^2dy

The derivative was found using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Now, we rewrite the integral in terms of u and solve:

$\frac{5}{3}\int\cos(u)du=\frac{5}{3}\sin(u)+C$

The integral was found using the following rule:

$\int \cos(x)dx=\sin(x)+C$

Finally, replace u with our original x term:

$\frac{5}{3}\sin(y^3)+C$

Report an Error

Example Question #953 : Ap Calculus Ab

Integrate:

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

Possible Answers:

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

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

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

$\frac{3}{2}e^{u}+C$

$\frac{1}{6}e^{x^2}+C$

Correct answer:

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

Explanation:

To integrate, we must perform the following substitution:

u=x^2, du=2xdx

The derivative was found using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Now, we rewrite the integral in terms of u and solve:

$\frac{3}{2}\int\ e^udu=\frac{3}{2}e^u+C$

The integral was found using the following rule:

$\int e^xdx =e^x+C$

Finally, replace u with our original x term:

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

Report an Error

View AP Calculus AB Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

View AP Calculus AB Tutors

Austin
Certified Tutor

University of Arizona, Bachelor of Science, Physiology.

View AP Calculus AB Tutors

Shakeel
Certified Tutor

All AP Calculus AB Resources

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

Popular Subjects

Reading Tutors in Philadelphia, MCAT Tutors in Philadelphia, Math Tutors in Denver, Computer Science Tutors in New York City, Statistics Tutors in Chicago, French Tutors in Phoenix, Math Tutors in Atlanta, English Tutors in San Francisco-Bay Area, English Tutors in Washington DC, MCAT Tutors in San Diego

Popular Courses & Classes

GMAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Chicago, GMAT Courses & Classes in Atlanta, SAT Courses & Classes in Seattle, LSAT Courses & Classes in Phoenix, SAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Los Angeles, MCAT Courses & Classes in Philadelphia, ISEE Courses & Classes in Boston, ACT Courses & Classes in New York City

Popular Test Prep

GRE Test Prep in San Diego, ISEE Test Prep in Denver, ACT Test Prep in Los Angeles, ACT Test Prep in San Diego, GMAT Test Prep in Miami, LSAT Test Prep in Denver, SAT Test Prep in New York City, LSAT Test Prep in Phoenix, ISEE Test Prep in Chicago, MCAT Test Prep in San Diego

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 #11 : Antiderivatives By Substitution Of Variables

Example Question #12 : Antiderivatives By Substitution Of Variables

Example Question #13 : Antiderivatives By Substitution Of Variables

Example Question #14 : Antiderivatives By Substitution Of Variables

Example Question #15 : Antiderivatives By Substitution Of Variables

Example Question #16 : Antiderivatives By Substitution Of Variables

Example Question #951 : Ap Calculus Ab

Example Question #952 : Ap Calculus Ab

Example Question #61 : Techniques Of Antidifferentiation

Example Question #953 : Ap Calculus Ab

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: