We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Indefinite 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 #61 : Indefinite Integrals

Integrate:

$\int x^2e^xdx$

Possible Answers:

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

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

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

e^x(x^2-2x+2)

Correct answer:

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

Explanation:

To solve the integral, we must use integration by parts, which states that

$\int udv=uv-\int vdu$

We must designate a u and a dv:

u=x^2, du=2xdx

dv=e^xdx, v=e^x

The derivative and integral of u and dv, respectively, were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\int e^xdx=e^x+C$

Using the above formula, we can rewrite the integral:

$\int x^2e^xdx=x^2e^x-\int 2xe^xdx$

We must use integration by parts again, because we have an integral that cannot be solved using other methods.

This time, using the same integration and derivation rules, we have

u=2x, du=2dx

dv=e^xdx, v=e^x

Using the same formula above, we get

$\int x^2e^xdx=x^2e^x-\int 2xe^xdx=x^2e^x-[2xe^x-\int 2e^xdx]$

Integrating, we get

$x^2e^x-[2xe^x-\int 2e^xdx]=x^2e^x-2xe^x+2e^x+C$

The same integration rule above was used.

Rewriting, our final answer is

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

Report an Error

Example Question #332 : Finding Integrals

Evaluate.

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

Possible Answers:

$F(x) = \frac{ \sin(2x)}{2} - \frac{\ln|2x|}{2}$

$F(x) = \frac{ \sin(x)}{2} - \frac{\ln(x)}{2}$

Answer not listed.

$F(x) = \sin(2x) - \ln(2x)$

$F(x) = \frac{ \sin(2x)}{4} + \frac{\ln(2x)}{2}$

Correct answer:

$F(x) = \frac{ \sin(2x)}{2} - \frac{\ln|2x|}{2}$

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

The antiderivative is $F(x) = \frac{ \sin(2x)}{2} - \frac{\ln|2x|}{2}$ .

Report an Error

Example Question #61 : Indefinite Integrals

Evaluate.

$\int \frac{1}{9x-8} \ dx$

Possible Answers:

Answer not listed.

$F(x) = \frac{\ln (9x-8)}{9}$

$F(x) = \ln (9x-8)$

$F(x) = \frac{\ln (9x-8)}{x}$

$F(x) = \frac{9x-8}{\ln (9x-8)}$

Correct answer:

$F(x) = \frac{\ln (9x-8)}{9}$

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, $f(x) = \frac{1}{9x-8}$ .

The antiderivative is $F(x) = \frac{\ln (9x-8)}{9}$ .

Report an Error

Example Question #61 : Indefinite Integrals

Evaluate.

$\int -\cos(7x) \ dx$

Possible Answers:

$F(x) = \frac{\cos(7x)}{7}$

Answer not listed.

$F(x) = - \sin(7x)$

$F(x) = \frac{\sin(x)}{7}$

$F(x) = - \frac{\sin(7x)}{7}$

Correct answer:

$F(x) = - \frac{\sin(7x)}{7}$

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, $f(x) =-\cos(7x)$ .

The antiderivative is $F(x) = - \frac{\sin(7x)}{7}$ .

Report an Error

Example Question #62 : Indefinite Integrals

Evaluate.

$\int e^{2x-5} - \sin(2x-5) \ dx$

Possible Answers:

$F(x) = \frac{e^{2x-5}}{2} + \frac{\cos(2x-5)}{2}$

$F(x) = \frac{e^{2x-5}}{x} - \frac{\cos(2x-5)}{x}$

Answer not listed.

$F(x) = e^{2x-5}+ \cos(2x-5)$

$F(x) = \frac{e^{2x-5}}{e^2} + \frac{\cos(2x-5)}{\cos2}$

Correct answer:

$F(x) = \frac{e^{2x-5}}{2} + \frac{\cos(2x-5)}{2}$

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, $f(x) =e^{2x-5} - \sin(2x-5)$ .

The antiderivative is $F(x) = \frac{e^{2x-5}}{2} + \frac{\cos(2x-5)}{2}$ .

Report an Error

Example Question #344 : Finding Integrals

Evaluate.

$\int 5x^4 + 12x \ dx$

Possible Answers:

F(x) = x^5 + 6x^2

$F(x) = \frac{1}{5}x^5 + \frac{1}{12}x^2$

F(x) = 5x^5 + 12x^2

Answer not listed.

F(x) = 5x^4 + 12x

Correct answer:

F(x) = x^5 + 6x^2

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) =5x^4 + 12x .

The antiderivative is F(x) = x^5 + 6x^2 .

Report an Error

Example Question #345 : Finding Integrals

Evaluate.

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

Possible Answers:

$F(x) = \frac{1}{ \ln(x-1)}$

$F(x) = 4 \ln(x-1)$

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

$F(x) = \ln(x-1)$

Answer not listed.

Correct answer:

$F(x) = 4 \ln(x-1)$

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, $f(x) = \frac{4}{x-1}$ .

The antiderivative is $F(x) = 4 \ln(x-1)$ .

Report an Error

Example Question #62 : Indefinite Integrals

Evaluate.

$\int 12 \ dx$

Possible Answers:

Answer not listed

F(x) =0

$F(x) = \frac{1}{12}x$

F(x) = 12x^2

F(x) = 12x

Correct answer:

F(x) = 12x

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 .

The antiderivative is F(x) = 12x .

Report an Error

Example Question #347 : Finding Integrals

Evaluate.

$\int -\sin(6x) +3 \ dx$

Possible Answers:

$F(x) = \frac{\cos(x)}{6x} + 3$

$F(x) = \frac{\cos(6x)}{6} + 3x$

$F(x) = -\cos(6x) + 3x$

Answer not listed.

$F(x) =- \frac{\cos(6x)}{6} + 3x$

Correct answer:

$F(x) = \frac{\cos(6x)}{6} + 3x$

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, $f(x) = -\sin(6x) +3$ .

The antiderivative is $F(x) = \frac{\cos(6x)}{6} + 3x$ .

Report an Error

Example Question #348 : Finding Integrals

Evaluate.

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

Possible Answers:

$F(x) = 2x^2 - \frac{\ln(3x)}{3}$

$F(x) = \frac{1}{2}x^2 - \ln(x)$

Answer not listed.

$F(x) = x^2 - \frac{\ln(x)}{3}$

$F(x) = 2x^2 - \ln(x)$

Correct answer:

$F(x) = x^2 - \frac{\ln(x)}{3}$

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, $f(x) = 2x - \frac{1}{3x}$ .

The antiderivative is $F(x) = x^2 - \frac{\ln(x)}{3}$ .

Report an Error

View Calculus 2 Tutors

Mohammad
Certified Tutor

Rutgers University-New Brunswick, Bachelor of Science, Mathematics. Rutgers University-New Brunswick, Master of Science, Stat...

View Calculus 2 Tutors

Jason
Certified Tutor

Brigham Young University, Bachelor, Neuroscience.

View Calculus 2 Tutors

Reza
Certified Tutor

University of Sherbrooke, Doctor of Philosophy, Mathematics. University of Manitoba, Master of Science, Mathematics.

All Calculus 2 Resources

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

Popular Subjects

LSAT Tutors in Seattle, Physics Tutors in Houston, Reading Tutors in Dallas Fort Worth, Algebra Tutors in New York City, ISEE Tutors in Seattle, MCAT Tutors in New York City, Computer Science Tutors in Denver, MCAT Tutors in Chicago, GMAT Tutors in Houston, Chemistry Tutors in Houston

Popular Courses & Classes

GRE Courses & Classes in Boston, ISEE Courses & Classes in Denver, LSAT Courses & Classes in Miami, ISEE Courses & Classes in Seattle, MCAT Courses & Classes in Denver, SAT Courses & Classes in San Diego, ISEE Courses & Classes in Boston, SAT Courses & Classes in Boston, GMAT Courses & Classes in New York City, MCAT Courses & Classes in Atlanta

Popular Test Prep

MCAT Test Prep in Washington DC, LSAT Test Prep in San Diego, LSAT Test Prep in Miami, ISEE Test Prep in Dallas Fort Worth, ACT Test Prep in Boston, MCAT Test Prep in Boston, LSAT Test Prep in Chicago, ISEE Test Prep in Seattle, GRE Test Prep in Seattle, ISEE Test Prep in Houston

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 : Indefinite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #61 : Indefinite Integrals

Example Question #332 : Finding Integrals

Example Question #61 : Indefinite Integrals

Example Question #61 : Indefinite Integrals

Example Question #62 : Indefinite Integrals

Example Question #344 : Finding Integrals

Example Question #345 : Finding Integrals

Example Question #62 : Indefinite Integrals

Example Question #347 : Finding Integrals

Example Question #348 : Finding Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: