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 #91 : Indefinite Integrals

Integrate:

$\int \frac{\ln x}{x^3}dx$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

To integrate, we must integrate by parts, which uses the following formula:

$\int u dv=uv-\int v du$

We must choose our u and dv, and differentiate and integrate, respectively:

$u=\ln(x)$ , $du=\frac{dx}{x}$

$dv= x^{-3}, v=-\frac{1}{2}x^{-2}$

The following rules were used:

$\frac{\mathrm{d} }{\mathrm{d} x} \ln x=\frac{1}{x}$ , $\int x^ndx=\frac{x^{n+1}}{n+1}+C$

Note that we do not include the constant of integration in this step.

Now, using the above formula, rewrite the integral:

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

After integrating, we get our final answer

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

using the same integration rule as above.

Report an Error

Example Question #92 : Indefinite Integrals

Integrate:

$\int \ln(\sqrt[4]{x})dx$

Possible Answers:

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

$\frac{1}{4}(x \ln x -x)$

$x \ln x -x+C$

$\frac{x \ln x}{4} -x+C$

Correct answer:

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

Explanation:

To integrate, we must integrate by parts, which is done using the following formula:

$\int udv=uv-\int vdu$

Before rewriting the integrand using integration by parts, we can use a logarithm property to rewrite it as

$\int \ln x^{\frac{1}{4}} dx=\frac{1}{4} \int \ln x dx$

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

$u=\ln x$ , $du= \frac{dx}{x}$

dv=dx , v=x

We used the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \ln x = \frac{1}{x}$ , $\int dx =x +C$

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

Using the above formula, we can rewrite the integral as

$\frac{1}{4}(x\ln x - \int x \cdot \frac{dx}{x})=\frac{1}{4}(x\ln x -\int dx)$

Integrating, we get

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

using the same rule as above.

Our final answer is

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

Report an Error

Example Question #93 : Indefinite Integrals

Integrate:

$\int \sin^2(\frac{3x}{4})dx$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

To integrate, we must rewrite the integrand using the half angle identity:

$\int \frac{1}{2}(1-\sin(\frac{3}{2}x))dx$

Now, integrate:

$\int \frac{1}{2}(1-\sin(\frac{3}{2}x))dx=\frac{1}{2}(x+\frac{1}{3}\cos(\frac{3}{2}x))=\frac{1}{2}x+\frac{1}{3}\cos(\frac{3}{2}x)+C$

We used the following rules to integrate:

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

Note that we combined the two constants of integration to make one C.

Report an Error

Example Question #371 : Finding Integrals

Evaluate.

$\int 12 + x \ dx$

Possible Answers:

F(x) = 12x + 2x +C

Answer not listed.

F(x) = 1 +C

$F(x) = \frac{6x}{2} + \frac{x^2}{2} +C$

$F(x) = 12x + \frac{x^2}{2} + C$

Correct answer:

$F(x) = 12x + \frac{x^2}{2} + C$

Explanation:

$\int f(x) \ dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, f(x) = 12 + x .

The antiderivative is $F(x) = 12x + \frac{x^2}{2} +C$ .

Report an Error

Example Question #95 : Indefinite Integrals

Evaluate.

$\int \frac{1}{2}x - 30x^{14} \ dx$

Possible Answers:

$F(x) = \frac{1}{x} - 15x +C$

$F(x) = \frac{x^2}{2} - 30x^{14} +C$

$F(x) = 4x^2- 14x^{13} +C$

Answer not listed.

$F(x) = \frac{x^2}{4} - 2x^{15} +C$

Correct answer:

$F(x) = \frac{x^2}{4} - 2x^{15} +C$

Explanation:

$\int f(x) \ dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, $f(x) = \frac{1}{2}x - 30x^{14}$ .

The antiderivative is $F(x) = \frac{x^2}{4} - 2x^{15} +C$ .

Report an Error

Example Question #96 : Indefinite Integrals

Evaluate.

$\int \frac{3}{x-5} \ dx$

Possible Answers:

$F(x) = \frac{\ln(x-5)}{3} +C$

$F(x) = \frac{3}{ln(x-5)} +C$

$F(x) = \frac{1}{3}\ln(x-5) +C$

Answer not listed.

$F(x) = \ln(x-5) +C$

Correct answer:

Answer not listed.

Explanation:

$\int f(x) \ dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, $f(x) = \frac{3}{x-5}$ .

The antiderivative is $F(x) = 3\ln(x-5) +C$ .

Report an Error

Example Question #97 : Indefinite Integrals

Evaluate.

$\int e^{7x} \ dx$

Possible Answers:

Answer not listed.

$F(x) = \frac{7}{e^{7x}}$

$F(x) =7e^{7x} +C$

$F(x) = \frac{e^{7x}}{7} +C$

$F(x) = e^{7x} +C$

Correct answer:

$F(x) = \frac{e^{7x}}{7} +C$

Explanation:

$\int f(x) \ dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, $f(x) = e^{7x}$ .

The antiderivative is $F(x) = \frac{e^{7x}}{7} +C$ .

Report an Error

Example Question #101 : Indefinite Integrals

Evaluate.

$\int \cos(6x) \ dx$

Possible Answers:

$F(x) =- \frac{\sin(6x)}{6} +C$

$F(x) =- 6\sin(6x) +C$

Answer not listed.

$F(x) = \frac{6}{\sin(6x)} +C$

$F(x) = \frac{\sin(6x)}{6} +C$

Correct answer:

$F(x) = \frac{\sin(6x)}{6} +C$

Explanation:

$\int f(x) \ dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, $f(x) = \cos(6x)$ .

The antiderivative is $F(x) = \frac{\sin(6x)}{6} +C$ .

Report an Error

Example Question #372 : Finding Integrals

Evaluate.

$\int \frac{x-4}{x} \ dx$

Possible Answers:

$F(x) = x\ln(x-4) +C$

Answer not listed.

$F(x) = x\ln(x) +C$

$F(x) =\frac{4}{x-4} +C$

$F(x) = x-4\ln(x) +C$

Correct answer:

$F(x) = x-4\ln(x) +C$

Explanation:

$\int f(x) \ dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, $f(x) = \frac{x-4}{x}$ .

The antiderivative is $F(x) = x-4\ln(x) +C$ .

Report an Error

Example Question #721 : Integrals

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

Possible Answers:

$cot^{-1}x+C$

$\frac{-1}{x}+C$

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

ln(x^2+1)+C

Correct answer:

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

Explanation:

This integral is a definition and should be committed to memory. These functions are extremely useful when integrating.

$\int \frac{1}{x^2+1}dx=tan^{-1}x+C.$

Remember, every time you have an indefinite integral, you need to add on a constant to the end.

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

MCAT Tutors in Atlanta, Biology Tutors in Phoenix, ACT Tutors in Houston, Algebra Tutors in Miami, MCAT Tutors in New York City, Statistics Tutors in San Francisco-Bay Area, ISEE Tutors in Washington DC, Biology Tutors in Denver, ACT Tutors in Atlanta, Algebra Tutors in San Diego

Popular Courses & Classes

ISEE Courses & Classes in Phoenix, MCAT Courses & Classes in Denver, ISEE Courses & Classes in New York City, GMAT Courses & Classes in Boston, GRE Courses & Classes in Washington DC, GRE Courses & Classes in Los Angeles, SAT Courses & Classes in Houston, GRE Courses & Classes in Denver, SSAT Courses & Classes in New York City, GMAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

GMAT Test Prep in Denver, ISEE Test Prep in Washington DC, SAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Atlanta, SAT Test Prep in Philadelphia, ACT Test Prep in Atlanta, GRE Test Prep in Los Angeles, GRE Test Prep in San Diego, ISEE Test Prep in Miami, GRE 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 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #91 : Indefinite Integrals

Example Question #92 : Indefinite Integrals

Example Question #93 : Indefinite Integrals

Example Question #371 : Finding Integrals

Example Question #95 : Indefinite Integrals

Example Question #96 : Indefinite Integrals

Example Question #97 : Indefinite Integrals

Example Question #101 : Indefinite Integrals

Example Question #372 : Finding Integrals

Example Question #721 : Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: