Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : Solving Integrals By Substitution

Solve the following indefinite integral:

$\int \frac{dx}{\sqrt{1-4x^2}}$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

To solve the following integral, we must make a substitution to create the following general form:

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

We make the following subsitution:

u=2x, du=2dx

The derivative was found using the following rule:

$\frac{d}{dx}(x^n)=nx^{n-1}$

The integral now looks like this:

$\frac{1}{2}\int \frac{du}{\sqrt{1-u^2}}$

Notice that it is in the same form as the integral we want.

Now use the form from above to integrate:

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

To finish the problem, replace u with 2x:

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

Report an Error

Example Question #4 : Solving Integrals By Substitution

Simplify the following indefinite integral.

$\small \int (\ln x)^2 dx$

Possible Answers:

$\small e^x(x^2-2x+2)+C$

$\small e^x((\ln x)^2-2\ln x+2)+C$

$\small \small \small x((\ln x )^2-2\ln x+2)+C$

$\small \small \small \small x((\ln x )^2+2\ln x-2)+C$

Correct answer:

$\small \small \small x((\ln x )^2-2\ln x+2)+C$

Explanation:

We can simplify

$\small \int (\ln x)^2 dx$

by first doing a substitution, with $\small u=\ln x$ , which gives us $\small e^u=x$ , which means that $\small e^udu=dx$ . So the integral becomes

$\small \small \int (\ln x)^2 dx=\int u^2e^u du$

The integral $\small \small \int u^2 e^u du$ can be solved using two integration by parts, which will give us

$\small \small \small \small \int u^2 e^u du=e^u(u^2-2u+2)+C$

So now we just plg in $\small \ln x$ into $\small u$ and $\small x$ into $\small e^u$ to get

$\small \small \small \small \int (\ln x)^2 dx=\int u^2e^u du=e^u(u^2-2u+2)+C=x((\ln x )^2-2\ln x+2)+C$

Report an Error

Example Question #9 : Solving Integrals By Substitution

Calculate the integral: $\int_{2}^{6}\frac{2}{3xln^3(x)} \:dx$

Possible Answers:

$\frac{1}{3ln^2(6)}-\frac{1}{3ln^2(2)}$

$\frac{1}{3ln^2(2)}-\frac{1}{3ln^2(6)}$

$\frac{2}{3ln^2(6)}-\frac{2}{3ln^2(2)}$

$\frac{2}{3ln^2(2)}-\frac{2}{3ln^2(6)}$

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

Correct answer:

$\frac{1}{3ln^2(2)}-\frac{1}{3ln^2(6)}$

Explanation:

Pull out the constant out in front of the integral.

$\int_{2}^{6}\frac{2}{3xln^3(x)} \:dx=\frac{2}{3}\int_{2}^{6}\frac{1}{xln^3(x)} \:dx$

Use U-substitution to solve.

u=ln(x)

$du=\frac{1}{x}\:dx$

$\frac{2}{3}\int_{2}^{6}\frac{1}{xln^3(x)} \:dx= \frac{2}{3}\int u^{-3}\:du= \frac{2}{3}\left[ \frac{u^{-2}}{-2}\right]=-\frac{1}{3}\left[\frac{1}{ln^2(u)}\right]_{2}^{6}$

$-\frac{1}{3}\left[\frac{1}{ln^2(6)}-\frac{1}{ln^2(2)}\right]=\frac{1}{3ln^2(2)}-\frac{1}{3ln^2(6)}$

Report an Error

Example Question #3 : Solving Integrals By Substitution

Solve the following indefinite integral:

$\int \frac{dx}{\sqrt{2x+3}}$

Possible Answers:

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

$\ln \left | \sqrt{2x+3} \right |+C$

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

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

Correct answer:

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

Explanation:

The integral can be solved with a clever substitution:

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

The derivative was found using the following rules:

$\frac{d}{dx}(x^n)=nx^{n-1}$ , $\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$

Then, when you rewrite the integral in terms of u, you find that you get:

$\int du = u+C$

The integration was performed using the following rule:

$\int dx=x+C$

Finally, replace the u with our original term.

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

Report an Error

Example Question #2691 : Calculus Ii

Solve the following indefinite integral:

$\int \frac{dx}{(x+2)^2+1}$

Possible Answers:

$\tan (x+2)+C$

$\arcsin (x+2)+C$

$\ln ((x+2)^2+1)+C$

$\arctan (x+2)+C$

Correct answer:

$\arctan (x+2)+C$

Explanation:

The integral is found by recognizing the following integral:

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

To solve the integral, make the following substitution:

u=x+2 , du=dx

The derivative was found using the following rule:

$\frac{d}{dx}(x^n)=nx^{n-1}$

The integral then becomes

$\int \frac{du}{u^2+1}=\arctan (u)+1$

Finish by replacing u with the original term containing x.

Report an Error

Example Question #2692 : Calculus Ii

Evaluate the following integral:

$\int{sin(x)cos(x)dx}$

Possible Answers:

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

$\frac{cos^2(x)sin^2(x)}{4}+c$

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

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

Correct answer:

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

Explanation:

Substitution can be used to make this problem easier. Let u = sin(x). Then du = cos(x)dx. This allows us to rewrite and evaluate the original integral in terms of u as:

$\int{udu}=\frac{u^2}{2}+c$

The final answer should be written in terms of the original variables, so substitute sin(x) for u.

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

Note that we could have also chosen cos(x) as u, but the above substitution avoids introducing negatives.

Report an Error

Example Question #11 : Solving Integrals By Substitution

Evaluate the integral $\int{\frac{1}{1+e^{2x}}dx}$ .

Possible Answers:

$e^{2x}+c$

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

arctan(e^x)+c

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

Correct answer:

arctan(e^x)+c

Explanation:

First, notice that $e^{2x}=(e^{x})^2$ . Use the substitution u=e^x, du=e^xdx to rewrite the integral as

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

Next, recall that $\int\frac{1}{1+x^2}dx=arctan(x)+c$ , so

$\int{\frac{1}{1+u^{2}}du}=arctan(u)+c$ .

Lastly, substitute e^x in place of to write the answer in terms of the original variables.

arctan(u)+c=arctan(e^x)+c

Report an Error

Example Question #2694 : Calculus Ii

Solve the following indefinite integral:

$\int \cos^4(x)\sin(x)dx$

Possible Answers:

$\sec^2(x)+C$

$\sin^2(x)+C$

$-\frac{\cos^5(x)}{5}+C$

$\frac{\sin^4(x)}{4}+C$

Correct answer:

$-\frac{\cos^5(x)}{5}+C$

Explanation:

To solve the indefinite integral, make a simple substitution:

$u=\cos(x), du=-\sin(x)dx$

The integral then becomes:

$\int -u^4du$

After integrating, we get

$-\frac{u^5}{5}+C$

The following rule was used for the integration:

$\int x^ndx=\frac{x^{n+1}}{n+1}+C$

Finally, replace the u with the original term containing x.

Report an Error

Example Question #2695 : Calculus Ii

Please solve the following integral:

$\int\frac{dx}{x(ln(x))^{2}}$

Possible Answers:

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

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

$\frac{1}{ln(x)} + C$

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

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

Correct answer:

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

Explanation:

We know that the derivative of ln(x) is $\frac{1}x{}$ .

Doing a substitution and setting

$u = \ln x$ and $du = \frac{dx}{x}$ allows us to rewrite the integral as

$\int \frac{du}{u^{2}}$ which can be rewritten as $\int u^{-2}du$ .

Integrating this gets you $\frac{-1}{u}$ plus a constant (which is stated in the original question that you can assume that we already have one). Substituting back in gives us the final answer, which is

$\frac{-1}{ln(x)} + C$ .

Report an Error

Example Question #12 : Solving Integrals By Substitution

Please solve the following integral.

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

Possible Answers:

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

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

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

$\arctan x + C$

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

Correct answer:

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

Explanation:

We know that the derivative of $\arctan(x)$ is $\frac{1}{1 + x^2}$ .

So substituting

$u = \arctan(x)$ allows us to have

$du= \frac{1}{1 + x^2}$ .

This allows us to rewrite the integral as

$\int u du$ which, when integrated, gives us

$\frac{u^2}{2} + C$ .

Substiting x back in gives us the answer,

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

Report an Error

View Calculus 2 Tutors

Samita
Certified Tutor

University of Houston, Bachelor of Science, Computer Science. University of Houston, Non Degree Bachelors, Biology, General.

View Calculus 2 Tutors

Christopher
Certified Tutor

Rose-Hulman Institute of Technology, Bachelor of Science, Physics. University of Washington, Master of Science, Physics.

View Calculus 2 Tutors

Muhammad
Certified Tutor

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

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Atlanta, Reading Tutors in Dallas Fort Worth, Statistics Tutors in Phoenix, Physics Tutors in Boston, Statistics Tutors in Denver, SAT Tutors in Boston, GMAT Tutors in Boston, Math Tutors in Miami, GMAT Tutors in Miami, SSAT Tutors in Los Angeles

Popular Courses & Classes

GRE Courses & Classes in Atlanta, Spanish Courses & Classes in Boston, ACT Courses & Classes in Miami, SSAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Miami, MCAT Courses & Classes in Philadelphia, GRE Courses & Classes in Phoenix, GMAT Courses & Classes in Chicago, SAT Courses & Classes in Chicago, GMAT Courses & Classes in New York City

Popular Test Prep

GMAT Test Prep in Houston, MCAT Test Prep in Atlanta, MCAT Test Prep in Miami, ISEE Test Prep in Seattle, ISEE Test Prep in Philadelphia, LSAT Test Prep in Miami, ACT Test Prep in Washington DC, SAT Test Prep in San Diego, SAT Test Prep in Boston, ACT Test Prep in Miami

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 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #2 : Solving Integrals By Substitution

Example Question #4 : Solving Integrals By Substitution

Example Question #9 : Solving Integrals By Substitution

Example Question #3 : Solving Integrals By Substitution

Example Question #2691 : Calculus Ii

Example Question #2692 : Calculus Ii

Example Question #11 : Solving Integrals By Substitution

Example Question #2694 : Calculus Ii

Example Question #2695 : Calculus Ii

Example Question #12 : Solving Integrals By Substitution

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: