We are open Saturday and Sunday!

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 #104 : Indefinite Integrals

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

We first need to simplify the expression by dividing the numerator by the denominator.

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

Now, we integrate each part. Since this is an indefinite integral, we need to add a constant to the ending.

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

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

Remember to rewrite the last term to make use of the power rule. $\frac{3}{x^2}=3x^{-2}.$

Report an Error

Example Question #105 : Indefinite Integrals

Integrate:

$\int 10t\sin(4t)dt$

Possible Answers:

$-\frac{5t}{2}\cos(4t)+\frac{5}{8}\sin(4t)+C$

$-\frac{5t}{2}\cos(4t)-\frac{5}{4}\sin(4t)+C$

$-\frac{5t}{2}\cos(4t)-\frac{5}{8}\sin(4t)$

$-\frac{5t}{2}\cos(4t)-\frac{5}{8}\sin(4t)+C$

Correct answer:

$-\frac{5t}{2}\cos(4t)-\frac{5}{8}\sin(4t)+C$

Explanation:

To integrate, we must use integration by parts, which is given by the following:

$\int udv=uv-\int vdu$

Now, we must designate our u and dv, and differentiate and integrate respectively:

u=10t , du=10dt

$dv=\sin(4t)$ , $v=-\frac{\cos(4t)}{4}$

Note that we don't include the constant of integration during this step.

The rules used are as follows:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\int \sin(x)dx=-\cos(x)+C$

Now, use the above formula to rewrite the integral:

$\frac{-5t\cos(4t)}{2}-\frac{5}{2}\int \cos(4t)dt$

Next, we integrate to get our final answer:

$-\frac{5t}{2}\cos(4t)-\frac{5}{8}\sin(4t)+C$

The integration was performed using the following rule:

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

Report an Error

Example Question #106 : Indefinite Integrals

$\int 4x^2-1dx$

Possible Answers:

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

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

4x^3-x+C

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

$\frac{4x^3}{3}-x$

Correct answer:

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

Explanation:

Remember that when integrating, raise the exponent by 1 and then put that result on the denominator. A constant yields an integral of x. Therefore, your answer is: $\frac{4x^3}{3}-x$ . Remember to add a +C because it is an indefinite integral: $\frac{4x^3}{3}-x+C$ .

Report an Error

Example Question #107 : Indefinite Integrals

$\int 4x^{\frac{1}{3}}-2xdx$

Possible Answers:

$x^{\frac{4}{3}}-x^2+C$

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

$3x^{\frac{4}{3}}-x^2+C$

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

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

Correct answer:

$3x^{\frac{4}{3}}-x^2+C$

Explanation:

To integrate, remember that you add one to the exponent and then also put that result on the denominator. After integrating, you get: $4(\frac{x^{\frac{4}{3}}}{\frac{4}{3}})-\frac{2x^2}{2}$ . Simplify and get $3x^{\frac{4}{3}}-x^2$ . Then, remember to add +C because it is an indefinite integral: $3x^{\frac{4}{3}}-x^2+C.$

Report an Error

Example Question #108 : Indefinite Integrals

$\int (4x-1)(2x+3)dx$

Possible Answers:

$\frac{8}{3}x^3+5x^2-3x$

x^3+5x^2-3x+C

$\frac{8}{3}x^3+5x^2-3x+C$

$\frac{8}{3}x^3-5x^2-3x+C$

$\frac{8}{3}x^3+5x^2-x+C$

Correct answer:

$\frac{8}{3}x^3+5x^2-3x+C$

Explanation:

First off, multiply the binomials using FOIL: $\int 8x^2+10x-3dx$ . Then, integrate, remembering to raise the exponent by 1 and then putting that result on the denominator: $\frac{8x^3}{3}+\frac{10x^2}{2}-3x$ . Simplify and add a +C because it is an indefinite integral: $\frac{8}{3}x^3+5x^2-3x+C$ .

Report an Error

Example Question #731 : Integrals

$\int_{1}^{3}\frac{4}{x}dx$

Possible Answers:

1.0264

4.3444

2.3944

4.3944

Correct answer:

4.3944

Explanation:

Recall that when integrating a single variable on the denominator, take ln of that term. Therefore, the integration is $4ln\left | x \right |$ . Then, evaluate at 3 and then 1. Subtract the results: 4ln3-4ln1=4.3944 .

Report an Error

Example Question #110 : Indefinite Integrals

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

Possible Answers:

$-\frac{x^4}{4}+\frac{2x^3}{3}+\frac{3x^2}{2}+6x+C$

$\frac{x^4}{4}+\frac{2x^3}{3}-\frac{3x^2}{2}+6x+C$

$-\frac{x^4}{4}+\frac{2x^3}{3}-\frac{3x^2}{2}+6x+C$

$-\frac{x^4}{4}+\frac{2x^3}{3}-\frac{3x^2}{2}+6x$

Correct answer:

$-\frac{x^4}{4}+\frac{2x^3}{3}-\frac{3x^2}{2}+6x+C$

Explanation:

The first step here should be multiplying the binomials:

(2-x)(3+x^2)=-x^3+2x^2-3x+6 .

Then, integrate. Remember to raise each power by one and then also put that result on the denominator.

Add a +C at the end because it is an indefinite integral.

Therefore, your answer should like this:

$-\frac{x^4}{4}+\frac{2x^3}{3}-\frac{3x^2}{2}+6x+C$ .

Report an Error

Example Question #383 : Finding Integrals

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

Possible Answers:

$\frac{7}{5}x^{\frac{5}{4}}+C$

$\frac{8}{5}x^{\frac{5}{4}}+C$

$\frac{8}{5}x^{\frac{5}{4}}$

$\frac{8}{5}x^{\frac{3}{4}}+C$

Correct answer:

$\frac{8}{5}x^{\frac{5}{4}}+C$

Explanation:

To integrate this, first pull the 2 outside of the integral sign:

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

Then, when integrating, remember to add 1 to the exponent and then also put that result on the denominator:

$2(\frac{x^{\frac{5}{4}}}{\frac{5}{4}})$ .

Then, simplify:

$2(\frac{4}{5})x^{\frac{5}{4}}=\frac{8}{5}x^\frac{5}{4}$ .

Then, add a +C because it is an indefinite integral:

$\frac{8}{5}x^{\frac{5}{4}}+C$ .

Report an Error

Example Question #384 : Finding Integrals

$\int \frac{x^3}{x^4-1}dx$

Possible Answers:

$\frac{1}{4}ln\left | x \right |+C$

$\frac{1}{4}ln\left | x^4-1 \right |+C$

$ln\left | x^4-1 \right |+C$

$\frac{1}{4}ln\left | x^4-1 \right |$

Correct answer:

$\frac{1}{4}ln\left | x^4-1 \right |+C$

Explanation:

Use u substitution here to integrate.

Assign

u=x^4-1 ,

$du=4x^3; \frac{1}{4}du=x^3$ .

Substitute in to rewrite the expression:

$\frac{1}{4}\int \frac{1}{u}du$ .

Recall that when integrating a single variable on the denominator, you take the natural log of that term.

Therefore, after integrating, you get

$\frac{1}{4}ln\left | u \right |$ .

Substitute in your initial expression to get

$\frac{1}{4}ln\left | x^4-1 \right |$ .

Remember to add a C because it is an indefinite integral:

$\frac{1}{4}ln\left | x^4-1 \right |+C$ .

Report an Error

Example Question #391 : Finding Integrals

$\int 9x^{\frac{1}{3}}dx$

Possible Answers:

$\frac{7}{4}x^{\frac{4}{3}}+C$

$9x^{\frac{4}{3}}+C$

$\frac{27}{4}x^{\frac{4}{3}}+C$

$\frac{27}{4}x^{\frac{3}{4}}+C$

Correct answer:

$\frac{27}{4}x^{\frac{4}{3}}+C$

Explanation:

First, put the 9 on the outside of the integral sign:

$9\int x^{\frac{1}{3}}dx$ .

Then, when integrating, remember to raise the exponent by 1 and put that result on the denominator:

$9(\frac{x\frac{4}{3}}{\frac{4}{3}})$ .

Simplify:

$9(\frac{3}{4})x^{\frac{4}{3}}=\frac{27}{4}x^{\frac{4}{3}}$ .

Remember to add a plus C at the end because it is an indefinite integral:

$\frac{27}{4}x^{\frac{4}{3}}+C.$

Report an Error

View Calculus 2 Tutors

Chisom
Certified Tutor

Georgia Southern University, Bachelor of Science, Biochemistry. University of Georgia, Current Grad Student, Pharmacy.

View Calculus 2 Tutors

Bahaa
Certified Tutor

Concordia University-Ann Arbor, Master of Arts Teaching, Mathematics Teacher Education.

View Calculus 2 Tutors

Christopher
Certified Tutor

Brigham Young University-Idaho, Bachelor of Science, Applied Mathematics.

All Calculus 2 Resources

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

Popular Subjects

Reading Tutors in Seattle, English Tutors in Phoenix, GRE Tutors in Seattle, Chemistry Tutors in San Diego, Computer Science Tutors in New York City, Statistics Tutors in Seattle, GRE Tutors in Miami, Algebra Tutors in Miami, Computer Science Tutors in Boston, ACT Tutors in Los Angeles

Popular Courses & Classes

ACT Courses & Classes in New York City, GMAT Courses & Classes in Houston, GRE Courses & Classes in Seattle, GRE Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Philadelphia, MCAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Atlanta, ISEE Courses & Classes in San Diego, GRE Courses & Classes in Atlanta

Popular Test Prep

GMAT Test Prep in Miami, LSAT Test Prep in Dallas Fort Worth, GMAT Test Prep in San Diego, SSAT Test Prep in Washington DC, LSAT Test Prep in San Diego, GRE Test Prep in Seattle, MCAT Test Prep in Washington DC, MCAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in San Diego, ISEE Test Prep in Denver

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 #104 : Indefinite Integrals

Example Question #105 : Indefinite Integrals

Example Question #106 : Indefinite Integrals

Example Question #107 : Indefinite Integrals

Example Question #108 : Indefinite Integrals

Example Question #731 : Integrals

Example Question #110 : Indefinite Integrals

Example Question #383 : Finding Integrals

Example Question #384 : Finding Integrals

Example Question #391 : Finding Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: