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 #761 : Integrals

Evaluate the integral:

$\int \frac{x^4+2x^2+5}{x}+\frac{1}{x^2+9}dx$

Possible Answers:

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\arctan(\frac{x}{3})+C$

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\frac{1}{3}\arcsin(\frac{x}{3})+C$

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\frac{1}{3}\arctan(\frac{x}{3})+C$

$\frac{x^3}{3}+x^2+5\ln\left | x\right |+\frac{1}{3}\arctan(\frac{x}{3})+C$

Correct answer:

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\frac{1}{3}\arctan(\frac{x}{3})+C$

Explanation:

To evaluate the integral, it is easiest if we split it into two integrals:

$\int \frac{x^4+2x^2+5}{x}dx+\int \frac{dx}{x^2+9}$

To integrate the first integral, we can divide to rewrite the integrand into something easier to integrate:

$\frac{x^4+2x^2+5}{x}=x^3+2x+\frac{5}{x}$

$\int (x^3+2x+\frac{5}{x})dx=\frac{x^4}{4}+x^2+5\ln\left | x\right |+C$

The following rules were used to integrate:

$\int x^n dx= \frac{x^{n+1}}{n+1}+C$ , $\int \frac{dx}{x}=\ln\left | x\right |+C$

Next, we integrate the second integral

$\int \frac{dx}{x^2+9}=\frac{1}{3}\arctan(\frac{x}{3})+C$

using the following rule:

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

Finally, combine the two results together:

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\frac{1}{3}\arctan(\frac{x}{3})+C$

Note that the constants of integration combine to make a single constant.

Report an Error

Example Question #762 : Integrals

Evaluate the following integral: $\int(2n^2+5n+1)dn$

Possible Answers:

$\frac{2n^3}{3}+\frac{5n^2}{2}+n+C$

2n^3+5n^2+n+C

$2n-\frac{5}{n}-n+C$

4n+5+C

Correct answer:

$\frac{2n^3}{3}+\frac{5n^2}{2}+n+C$

Explanation:

To evaluate the integral, we use the following rule: $\int(x^n)dx=\frac{x^{n+1}}{n+1}+C$ . Applying the rule, we get $\int(2n^2+5n+1)dn$ = $\frac{2n^3}{3}+\frac{5n^2}{2}+n+C$

Report an Error

Example Question #763 : Integrals

Evaluate the following integral: $\int (\sin(x)+3x^2)dx$

Possible Answers:

$-\cos(x)+6x+C$

$\cos(x)+x^3+C$

$-\cos(x)+x^3+C$

$\sin(x)+x^3+C$

Correct answer:

$-\cos(x)+x^3+C$

Explanation:

Using the rules for integration, $\int x^ndx=\frac{x^{n+1}}{n+1}+C$ and $\int \sin(x)= -\cos(x)+C$ , the integral is evaluated. For the first part, we established the value, and for the term involving the exponent, $\int 3x^2dx=3x^3/3=x^3+C$ . We put these two things together to get the correct answer.

Report an Error

Example Question #764 : Integrals

Evaluate.

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

Possible Answers:

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

F(x) = 4 x^2 + C

Answer not listed.

F(x) = 2 x^2 + C

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

Correct answer:

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

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, $f(x) = \frac{1}{2} x$ .

The antiderivative is $F(x) = \frac{1}{4} x^2 + C$ .

Report an Error

Example Question #765 : Integrals

Evaluate.

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

Possible Answers:

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

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

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

Answer not listed.

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

Correct answer:

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

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, $f(x) = e^ {x -7}$ .

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

Report an Error

Example Question #766 : Integrals

Evaluate.

$\int \frac{x-4}{x-5} \ dx$

Possible Answers:

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

Answer not listed.

$F(x) =5\ln(x-4) + x + C$

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

$F(x) =\ln(x-5)(x-4) + C$

Correct answer:

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

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, $f(x) =\frac{x-4}{x-5}$ .

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

Report an Error

Example Question #767 : Integrals

$\int \frac{2t-1}{t}dt$

Possible Answers:

$2t^2-ln\left | t \right |+C$

$2t-ln\left | t \right |+C$

$2t-ln\left | t \right |$

$2t-2ln\left | t \right |+C$

$t-ln\left | t \right |+C$

Correct answer:

$2t-ln\left | t \right |+C$

Explanation:

First, chop up the fraction into two separate terms:

$\int 2-\frac{1}{t}dt$

Then, integrate.

$2t-ln\left | t \right |$

Now, remember to add a C because it is an indefinite integral:

$2t-ln\left | t \right |+C$ .

Report an Error

Example Question #141 : Indefinite Integrals

$\int 5t^2-2t+6dt$

Possible Answers:

$\frac{5}{3}t^3+t^2+6t+C$

$\frac{5}{3}t^3-t^2+6t+C$

$\frac{5}{3}t^3-t^2-6t+C$

$\frac{5}{3}t^3-t^2+6t$

$\frac{1}{3}t^3-t^2+6t+C$

Correct answer:

$\frac{5}{3}t^3-t^2+6t+C$

Explanation:

Integrate this expression. Remember to add one to the exponent and also put that result on the denominator:

$5(\frac{t^3}{3})-2(\frac{t^2}{2})+6t$

Simplify and add C because it is an indefinite integral:

$\frac{5}{3}t^3-t^2+6t+C$ .

Report an Error

Example Question #142 : Indefinite Integrals

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

Possible Answers:

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

$\frac{3}{7}x^{\frac{1}{3}}+C$

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

$\frac{3}{8}x^{\frac{7}{3}}+C$

$\frac{2}{7}x^{\frac{7}{3}}+C$

Correct answer:

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

Explanation:

To integrate this expression, remember to raise the exponent by 1 and then also put that result on the denominator:

$\frac{x^{\frac{7}{3}}}{\frac{7}{3}}$

Simplify and add C because it is an indefinite integral:

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

Report an Error

Example Question #143 : Indefinite Integrals

$\int \frac{4x^3}{x^4+3}dx$

Possible Answers:

$ln\left | 4x^3 \right |+C$

$ln\left | x^4-3 \right |+C$

$ln\left | x^4+3 \right |+C$

$2ln\left | x^4+3 \right |+C$

$ln\left | x^4+3 \right |$

Correct answer:

$ln\left | x^4+3 \right |+C$

Explanation:

First, you must use u substitution to integrate this expression:

u=x^4+3; du=4x^3dx

Now, plug back in so you can integrate:

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

Now integrate:

$ln\left | u \right |$

Now substitute back in your initial expression and add C because it is an indefinite integral:

$ln\left | x^4+3 \right |+C$

Report an Error

View Calculus 2 Tutors

John
Certified Tutor

Oregon State University, Bachelor of Science, Engineering Physics. Oregon Health Science University, Master of Science, Biom...

View Calculus 2 Tutors

Riley
Certified Tutor

Reed College, Bachelor in Arts, Mathematics.

View Calculus 2 Tutors

Jerry
Certified Tutor

Worcester Polytechnic Institute, Bachelor of Science, Mechanical Engineering. University of Rochester, Master of Science, App...

All Calculus 2 Resources

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

Popular Subjects

Computer Science Tutors in Atlanta, Computer Science Tutors in Washington DC, English Tutors in Atlanta, Computer Science Tutors in San Francisco-Bay Area, Reading Tutors in Miami, MCAT Tutors in New York City, GMAT Tutors in Seattle, LSAT Tutors in Boston, Math Tutors in Dallas Fort Worth, Chemistry Tutors in Houston

Popular Courses & Classes

SSAT Courses & Classes in Atlanta, ISEE Courses & Classes in Atlanta, ISEE Courses & Classes in Miami, ACT Courses & Classes in Denver, MCAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Boston, ACT Courses & Classes in Phoenix, GRE Courses & Classes in Los Angeles, ACT Courses & Classes in Chicago, SSAT Courses & Classes in Washington DC

Popular Test Prep

GMAT Test Prep in Chicago, LSAT Test Prep in Philadelphia, ISEE Test Prep in Washington DC, ACT Test Prep in Chicago, LSAT Test Prep in Boston, MCAT Test Prep in Dallas Fort Worth, SAT Test Prep in Chicago, SSAT Test Prep in Boston, GRE Test Prep in Denver, LSAT Test Prep in New York City

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 #761 : Integrals

Example Question #762 : Integrals

Example Question #763 : Integrals

Example Question #764 : Integrals

Example Question #765 : Integrals

Example Question #766 : Integrals

Example Question #767 : Integrals

Example Question #141 : Indefinite Integrals

Example Question #142 : Indefinite Integrals

Example Question #143 : Indefinite Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: