Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Finding 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 #1 : Finding Integrals

Evaluate:

$\int_{-1}^{1} x ^{888}\; \mathrm{d} x$

Possible Answers:

$\frac{ 2}{889}$

$\frac{ 1}{444}$

$\frac{ 1}{888}$

$\frac{ 1}{889}$

Correct answer:

$\frac{ 2}{889}$

Explanation:

$\int x ^{n} \;\mathrm{d} x = \frac{ x ^{n+ 1 }}{n+1}$ ,

$\int_{-1}^{1} x ^{888}\; \mathrm{d} x$

$=\frac{ x ^{888+ 1 } }{888+ 1} \left| \begin{matrix} 1 \\ -1 \end{matrix}\right.$

$=\frac{ x ^{889} }{889} \left| \begin{matrix} 1 \\ -1 \end{matrix}\right.$

$=\frac{ 1 ^{889} }{889} - \frac{ (-1) ^{889} }{889} = \frac{ 1 }{889} - \frac{ -1 }{889} = \frac{ 1 }{889} +\frac{ 1 }{889} = \frac{ 2}{889}$

Report an Error

Example Question #1 : Finding Integrals

Evaluate:

$\int_{0}^{1} \ln (x^2 + 4x + 4) \; \mathrm{d}x$

Possible Answers:

$= -2 + \ln \frac{27}{4}$

The integral is undefined.

$= -1 + \ln \frac{729}{16}$

$= -2 + \ln \frac{729}{16}$

$= -1 + \ln \frac{27}{4}$

Correct answer:

$= -2 + \ln \frac{729}{16}$

Explanation:

Rewrite this as follows:

$\int_{0}^{1} \ln (x^2 + 4x + 4) \; \mathrm{d}x$

$= \int_{0}^{1} \ln (x +2)^2 \; \mathrm{d}x$

$= \int_{0}^{1} 2 \ln (x +2) \; \mathrm{d}x$

$= 2 \int_{0}^{1} \ln (x +2) \; \mathrm{d}x$

Substitute u = x + 2 . Then $\frac{\mathrm{d} u}{\mathrm{d} x} = 1$ and $\mathrm{d}x = \mathrm{d}x$ , and the bounds of integration become 2 and 3, making the integral equal to

$= 2 \int_{2}^{3} \ln u \; \mathrm{d}u$

$= \left.\begin{matrix} 2 \left (u \ln u - u \right ) \\ \\ \end{matrix}\right| \begin{matrix} 3\\ \\ 2 \end{matrix}$

$= 2 [(3 \ln 3 - 3 ) - (2 \ln 2 - 2)]$

$= 2 [ 3 \ln 3 - 2 \ln 2 - 1 ]$

$= 6 \ln 3 - 4 \ln 2 - 2$

$= \ln 3^6 - \ln 2^4 - 2$

$= \ln 729 - \ln 16 - 2$

$= -2 + \ln \frac{729}{16}$

Report an Error

Example Question #2 : Finding Integrals

Evaluate:

$\int_{4}^{16} \log_{4} x\; \mathrm{d}x$

Possible Answers:

$36- 6 \ln 2$

$36- \frac{ 6 }{\ln 2}$

$28- 6 \ln 2$

$28- \frac{ 6 }{\ln 2}$

The integral is undefined.

Correct answer:

$28- \frac{ 6 }{\ln 2}$

Explanation:

$\int_{4}^{16} \log_{4} x\; \mathrm{d}x$

$= \int_{4}^{16} \frac{\ln x}{\ln 4}\; \mathrm{d}x$

$= \frac{1}{\ln 4} \int_{4}^{16}\ln x \; \mathrm{d}x$

$= \frac{1}{\ln 4} \cdot \left.\begin{matrix} \left (x \ln x - x \right ) \\ \\ \end{matrix}\right| \begin{matrix} 16\\ \\ 4 \end{matrix}$

$= \frac{1}{\ln 4} \cdot \left [ \left (16 \ln 16 - 16 \right ) - \left (4 \ln 4 - 4 \right ) \right ]$

$= \frac{1}{\ln 4} \cdot \left (16 \ln 16 - 4 \ln 4 - 12 \right )$

$= \frac{ 16 \ln 16 }{\ln 4} - \frac{ 4 \ln 4 }{\ln 4} - \frac{ 12 }{\ln 4}$

$= 16 \log_{4} 16 - 4 - \frac{ 12 }{\ln 4} = 16 \cdot 2 - 4 - \frac{ 12 }{2 \ln 2} = 32 - 4 - \frac{ 6 }{\ln 2} = 28- \frac{ 6 }{\ln 2}$

Report an Error

Example Question #1 : Finding Integrals

Evaluate:

$\int_{1}^{ e/2} \ln 2x \; \mathrm{d}x$

Possible Answers:

$\frac{1 }{2}+ \ln 2$

$1 + \ln 2$

$1 - \ln 2$

$\frac{1 }{2}- \ln 2$

$\ln 2$

Correct answer:

$1 - \ln 2$

Explanation:

$\int_{1}^{ e/2} \ln 2x \; \mathrm{d}x$

Substitute u = 2x ; so $\frac{\mathrm{d} u}{\mathrm{d} x} = 2$ and $\; \mathrm{d} x = \frac{1}{2}\mathrm{d} u$ , and the bounds of integration become 2 and ; the above becomes

$\frac{1}{2}\int_{2}^{ e} \ln u \; \mathrm{d}u$

$= \left.\begin{matrix} \frac{1}{2} \cdot u \ln u - u\\ \\ \end{matrix}\right| \begin{matrix} e\\ \\ 2 \end{matrix}$

$= \frac{1}{2} \cdot \left [ \left ( e \ln e - e \right) - \left ( 2 \ln 2 - 2 \right) \right]$

$= \frac{1}{2} \cdot \left [ \left ( e \cdot 1 - e \right) - \left ( 2 \ln 2 - 2 \right) \right]$

$= \frac{1}{2} \cdot \left [ \left ( e - e \right) - \left ( 2 \ln 2 - 2 \right) \right] = \frac{1}{2} \cdot \left [2 - 2 \ln 2 \right] = 1 - \ln 2$

Report an Error

Example Question #2 : Finding Integrals

Which of the following functions makes the statement $\int_{0}^{3} f(x) \; \mathrm{d}x = 0$ true?

Possible Answers:

$f(x) = \sin \frac{3 \pi x}{4}$

$f(x) = \sin \frac{ \pi x}{3}$

$f(x) = \sin \frac{2 \pi x}{3}$

$f(x) = \sin \frac{3 \pi x}{2}$

$f(x) = \sin 3 \pi x$

Correct answer:

$f(x) = \sin \frac{2 \pi x}{3}$

Explanation:

$\int_{0}^{3} \sin nx \; \mathrm{d}x = \left.\begin{matrix} -\frac{\cos nx}{n}\\ \\ \end{matrix}\right|\begin{matrix} 3\\ \\ 0 \end{matrix} = -\frac{\cos 3n }{n} + \frac{\cos 0}{n} = \frac{1-\cos 3n }{n}$

Therefore, we are looking for a value of for which

$\int_{0}^{3} \sin nx \; \mathrm{d}x = \frac{1-\cos 3n }{n} = 0$ , or, equivalently,

$1-\cos 3n = 0$ , or

$\cos 3n = 1$

$3n = 0, 2 \pi, 4 \pi, 6\pi ...$

$n = 0, \frac{2 \pi}{3}, \frac{4 \pi}{3}, 2\pi ...$

The only choice that makes an element of this set is

$f(x) = \sin \frac{2 \pi x}{3}$ .

Report an Error

Example Question #2 : Finding Integrals

Evaluate:

$\int_{0}^{4 \pi} | \cos x | \; \mathrm{d}x$

Possible Answers:

$8 \sqrt{2}$

$4 \sqrt{2}$

$4 \sqrt{3}$

Correct answer:

Explanation:

An easy way to look at this is to note that on the interval $[0, 4 \pi]$ , the integrand

$f(x)=| \cos x |$

can be rewritten as

$f(x)= \begin{Bmatrix} \cos x & 0 \leq x \leq\frac{\pi}{2} \\ \\ - \cos x & \frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \\ \\ \cos x & \frac{3\pi}{2} \leq x \leq \frac{5\pi}{2} \\ \\ - \cos x & \frac{5\pi}{2} \leq x \leq \frac{7\pi}{2} \\ \\ \cos x & \frac{7\pi}{2} \leq x \leq 4\pi \end{matrix}$

Therefore,

$\int_{0}^{4 \pi} | \cos x | \; \mathrm{d}x$

$= \int_{0}^{4 \pi} f(x) \; \mathrm{d}x$

$= \int_{0}^{\frac{\pi}{2}} f(x) \; \mathrm{d}x + \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} f(x) \; \mathrm{d}x + \int_{\frac{3\pi}{2}}^{\frac{5\pi}{2}} f(x) \; \mathrm{d}x + \int_{\frac{5\pi}{2}}^{\frac{7\pi}{2}} f(x) \; \mathrm{d}x + \int_{\frac{7\pi}{2}}^{4 \pi} f(x) \; \mathrm{d}x$

$= \int_{0}^{\frac{\pi}{2}} \cos x \; \mathrm{d}x - \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos x \; \mathrm{d}x + \int_{\frac{3\pi}{2}}^{\frac{5\pi}{2}} \cos x \; \mathrm{d}x - \int_{\frac{5\pi}{2}}^{\frac{7\pi}{2}} \cos x \; \mathrm{d}x + \int_{\frac{7\pi}{2}}^{4 \pi} \cos x \; \mathrm{d}x$

The antiderivative of $g(x)= \cos x$ is $G(x) = \sin x$ . We can evaluate at each boundary of integration:

$G (0) = \sin 0 = 0$

$G\left ( \frac{\pi }{2}\right ) = \sin \frac{\pi }{2} = 1$

$G\left ( \frac{3\pi }{2}\right ) = \sin \frac{3\pi }{2} = -1$

$G\left ( \frac{5\pi }{2}\right ) = \sin \frac{5\pi }{2} = 1$

$G\left ( \frac{7\pi }{2}\right ) = \sin \frac{7\pi }{2} = -1$

$G (4\pi) = \sin 4\pi = 0$

Then

$\int_{0}^{\frac{\pi}{2}} \cos x \; \mathrm{d}x = \left ( G \left ( \frac{\pi }{2}\right ) - G (0)\right ) = 1 - 0= 1$

$\int_{\frac{\pi}{2}}^{ \frac{3\pi}{2}} \cos x \; \mathrm{d}x = \left ( G \left ( \frac{3\pi }{2}\right ) - G \left ( \frac{\pi }{2}\right )\right ) = -1 -1= -2$

$\int_{\frac{3\pi}{2}}^{ \frac{5\pi}{2}} \cos x \; \mathrm{d}x = \left ( G \left ( \frac{5\pi }{2}\right ) - G \left ( \frac{3\pi }{2}\right )\right ) = 1 - (-1)= 2$

$\int_{\frac{5\pi}{2}}^{ \frac{7\pi}{2}} \cos x \; \mathrm{d}x = \left ( G \left ( \frac{7\pi }{2}\right ) - G \left ( \frac{5\pi }{2}\right )\right ) = -1 -1= -2$

$\int_{\frac{7\pi}{2}}^{4 \pi} \cos x \; \mathrm{d}x = \left ( G (4\pi) -G \left ( \frac{7\pi }{2}\right ) \right ) = 0 - (-1) = 1$

The original integral can be evaluated as

$\int_{0}^{4 \pi} | \cos x | \; \mathrm{d}x = 1 - (-2) + 2 - (-2) + 1 = 1 + 2 + 2 + 2 + 1 = 8$

Report an Error

Example Question #1 : Finding Integrals

Evaluate:

$\int_{0}^{1}\int_{0}^{\pi} y^{2}\cos x \; \mathrm{d} x \; \mathrm{d} y$

Possible Answers:

$\frac{\pi}{2}$

$\frac{\pi}{3}$

$\frac{\pi \sqrt{3}}{2}$

$\frac{\pi \sqrt{2}}{2}$

Correct answer:

Explanation:

$\int_{0}^{1}\int_{0}^{\pi} y^{2}\cos x \; \mathrm{d} x \; \mathrm{d} y$

$=\int_{0}^{1}y^{2}\left (\int_{0}^{\pi}\cos x \; \mathrm{d} x \right )\; \mathrm{d} y$

We evaluate

$\int_{0}^{\pi}\cos x \; \mathrm{d} x$

$= \left.\begin{matrix} \sin x\\ \\ \end{matrix}\right|\begin{matrix} \pi \\ \\ 0 \end{matrix}$

$= \sin \pi - \sin 0 = 0 - 0 = 0$

The original double integral is now

$=\int_{0}^{1}y^{2} \cdot 0 \; \mathrm{d} y = \int_{0}^{1} 0 \; \mathrm{d} y= 0$

Report an Error

Example Question #2 : Finding Integrals

Evaluate:

$\int_{0}^{1}\int_{0}^{\pi} y^{2}\sin x \; \mathrm{d} x \; \mathrm{d} y$

Possible Answers:

$\frac{ \sqrt{3}}{3}$

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{2\sqrt{3}}{3}$

Correct answer:

$\frac{2}{3}$

Explanation:

$\int_{0}^{1}\int_{0}^{\pi} y^{2}\sin x \; \mathrm{d} x \; \mathrm{d} y$

$=\int_{0}^{1}y^{2}\left (\int_{0}^{\pi}\sin x \; \mathrm{d} x \right )\; \mathrm{d} y$

We evaluate

$\int_{0}^{\pi}\sin x \; \mathrm{d} x$

$= \left.\begin{matrix} -\cos x\\ \\ \end{matrix}\right|\begin{matrix} \pi \\ \\ 0 \end{matrix}$

$=- \cos \pi - (- \cos 0) = - (-1) - (-1) = 1 +1 = 2$

The original double integral is now

$=\int_{0}^{1}y^{2} \cdot 2 \; \mathrm{d} y$

$= 2 \int_{0}^{1}y^{2} \; \mathrm{d} y$

$= 2 \cdot \left.\begin{matrix} \frac{y^{3}}{3}\\ \\ \end{matrix}\right|\begin{matrix} 1 \\ \\ 0 \end{matrix}$

$=2 \cdot \left ( \frac{1^{3}}{3} - \frac{0^{3}}{3} \right ) =2 \cdot \left ( \frac{1 }{3} - \frac{0 }{3 \right )} = \frac{2 }{3}$

Report an Error

Example Question #5 : Finding Integrals

Evaluate:

$\int_{0}^{1}\int_{0}^{\ln 2} e^{x}y^{2} \; \mathrm{d} x \; \mathrm{d} y$

Possible Answers:

$\frac{4}{3}$

$\frac{2}{3}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

$\int_{0}^{1}\int_{0}^{\ln 2} e^{x}y^{2} \; \mathrm{d} x \; \mathrm{d} y$

$= \int_{0}^{1}y^{2} \left (\int_{0}^{\ln 2} e^{x} \; \mathrm{d}x \right ) \; \mathrm{d} y$

We evaluate

$\int_{0}^{\ln 2} e^{x} \; \mathrm{d}x$

$= \left.\begin{matrix} e^{x}\\ \\ \end{matrix}\right|\begin{matrix} \ln 2 \\ \\ 0 \end{matrix}$

$= e^{ln 2 } - e^{0} = 2 - 1 = 1$

The original double integral is now

$\int_{0}^{1}y^{2} \cdot 1 \; \mathrm{d} y$

$=\int_{0}^{1}y^{2} \; \mathrm{d} y$

$= \left.\begin{matrix} \frac{y^{3}}{3} \\ \\ \end{matrix}\right|\begin{matrix} 1 \\ \\ 0 \end{matrix}$

$= \frac{1^{3}}{3} - \frac{0^{3}}{3} = \frac{1}{3} - \frac{0}{3} = \frac{1}{3}$

Report an Error

Example Question #6 : Finding Integrals

Evaluate:

$\int_{0}^{\ln 2}\int_{0}^{\ln 2} e^{x+y} \; \mathrm{d} x \; \mathrm{d} y$

Possible Answers:

e^2

Correct answer:

Explanation:

The problem is easier if it is written as follows:

$\int_{0}^{\ln 2}\int_{0}^{\ln 2} e^{x+y} \; \mathrm{d} x \; \mathrm{d} y$

$= \int_{0}^{\ln 2}\int_{0}^{\ln 2} e^{x} e^{y} \; \mathrm{d} x \; \mathrm{d} y$

$= \int_{0}^{\ln 2}e^{y} \left (\int_{0}^{\ln 2} e^{x} \; \mathrm{d} x \right ) \; \mathrm{d} y$

We evaluate

$\int_{0}^{\ln 2} e^{x} \; \mathrm{d}x$

$= \left.\begin{matrix} e^{x}\\ \\ \end{matrix}\right|\begin{matrix} \ln 2 \\ \\ 0 \end{matrix}$

$= e^{ln 2 } - e^{0} = 2 - 1 = 1$

The original double integral is now

$= \int_{0}^{\ln 2}e^{y} \cdot 1 \; \mathrm{d} y$

$= \int_{0}^{\ln 2}e^{y} \; \mathrm{d} y$

which, similarly to $\int_{0}^{\ln 2} e^{x} \; \mathrm{d}x$ , is equal to 1.

Report an Error

View Calculus 2 Tutors

Edem
Certified Tutor

University of Ghana, Bachelor, Mathematics. African Institute of Mathematical Science, Master's/Graduate, Mathematics.

View Calculus 2 Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

View Calculus 2 Tutors

Samita
Certified Tutor

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

All Calculus 2 Resources

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

Popular Subjects

Reading Tutors in Chicago, GMAT Tutors in San Francisco-Bay Area, Calculus Tutors in Miami, Algebra Tutors in Dallas Fort Worth, SSAT Tutors in Denver, Statistics Tutors in Washington DC, English Tutors in Chicago, English Tutors in Seattle, Algebra Tutors in Seattle, Biology Tutors in Denver

Popular Courses & Classes

ACT Courses & Classes in Boston, SAT Courses & Classes in San Diego, Spanish Courses & Classes in Denver, MCAT Courses & Classes in Los Angeles, SSAT Courses & Classes in Miami, Spanish Courses & Classes in Los Angeles, GMAT Courses & Classes in Philadelphia, ISEE Courses & Classes in Washington DC, SAT Courses & Classes in Miami, ACT Courses & Classes in Seattle

Popular Test Prep

ACT Test Prep in Boston, ACT Test Prep in San Diego, LSAT Test Prep in Boston, ACT Test Prep in Atlanta, GMAT Test Prep in Philadelphia, ISEE Test Prep in Washington DC, GMAT Test Prep in Miami, SAT Test Prep in San Francisco-Bay Area, ACT Test Prep in Seattle, LSAT Test Prep in San Francisco-Bay Area

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 : Finding Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1 : Finding Integrals

Example Question #1 : Finding Integrals

Example Question #2 : Finding Integrals

Example Question #1 : Finding Integrals

Example Question #2 : Finding Integrals

Example Question #2 : Finding Integrals

Example Question #1 : Finding Integrals

Example Question #2 : Finding Integrals

Example Question #5 : Finding Integrals

Example Question #6 : Finding Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: