Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Definite 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 #31 : Finding Integrals

Evaluate the indefinite integral $\int x^3+\pi^2dx$ .

Possible Answers:

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

$x^3+\pi^2+C$

$4x^3+3\pi^2+C$

None of the other answers

Correct answer:

None of the other answers

Explanation:

The correct answer is $\frac{x^4}{4}+\pi^2x+C$ .

The integral itself is not too difficult to take, simply use the Power Rule on the x^3 and $\pi^2$ terms. The trick is to be careful when integrating $\pi^2$ . $\pi^2$ is a constant value (about 9.869 ) not a variable, so it must be integrated accordingly.

Report an Error

Example Question #22 : Definite Integrals

Evaluate $\int^{1}_{0} 5\tan^{-1}(x)dx$ .

Possible Answers:

$\frac{\pi+\ln2}{2}$

None of the other answers

$\frac{\pi^2-2}{\ln(5)}$

$\frac{5\pi-10\ln2}{4}$

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

Correct answer:

$\frac{5\pi-10\ln2}{4}$

Explanation:

This integral requires integration by parts followed by u-substitution. Here are the details

$\int^{1}_{0} 5\tan^{-1}(x)dx$ . Start

$=5\int^{1}_{0} \tan^{-1}(x)dx$ . Factor out the

Set up integration by parts with $u = \tan^{-1}(x)$ , dv = dx . We then have $du = \frac{1}{1+x^2}dx$ and v=x . Afterward, we use the integration by parts formula $uv - \int vdu$ .

$= 5\{[x\tan^{-1}(x)]^{1}_{0}-\int^{1}_{0} \frac{x}{1+x^2}dx\}$ .

Now at this point we use u-substitution to evaluate the 2nd integral. Let u =1+x^2 , then du = 2xdx, and therefore $\frac{1}{2}du = xdx$ . Substituting into the integral we have

$= 5\{[x\tan^{-1}(x)]^{1}_{0}-\frac{1}{2}\int^{2}_{1} \frac{1}{u}du\}$ . (Don't forget to change the bounds of integration by plugging them into for our equation for .)

$= 5\{[x\tan^{-1}(x)]^{1}_{0}-[\frac{1}{2}\ln(u)]^{2}_{1}\}$

$=5((\frac{\pi}{4}) - (\frac{1}{2}\ln2))$

$= \frac{5\pi-10\ln2}{4}$

Report an Error

Example Question #31 : Finding Integrals

Evaluate $\int^{1}_{-1}xe^{x^4}dx$ .

Possible Answers:

$\ln2-\frac{1}{2}$

Not possible without a calculator

$\frac{1}{2}$

Correct answer:

Explanation:

This integral isn't possible to integrate directly using antiderivatives, but we can still find its value by noticing that $xe^{x^4}$ is an odd function (f(-x)=-f(x)) , and that our limits of integration are negatives of each other.

Hence

$\int^{1}_{-1}xe^{x^4}dx = \int^{1}_{0}xe^{x^4}dx +\int^{0}_{-1}xe^{x^4}dx$

$=\int^{1}_{0}xe^{x^4}dx - \int^{-1}_{0}xe^{x^4}dx$

$=\int^{1}_{0}xe^{x^4}dx - \int^{1}_{0}xe^{x^4}dx$ . (Since $xe^{x^4}$ is an odd function)

= 0

Report an Error

Example Question #21 : Definite Integrals

Evaluate $\int_{0}^{\pi/2} \sin^{99}(x)\cos(x)dx$

Possible Answers:

$\frac{\pi^{99}}{99}$

$\frac{1}{100}$

None of the other answers.

$\frac{\pi^2}{2}$

Correct answer:

$\frac{1}{100}$

Explanation:

We can use u-substitution for this integral

$\int_{0}^{\pi/2} \sin^{99}(x)\cos(x)dx$ . Start

Let $u= \sin(x)$ , then $du = \cos(x)dx$ , and our integral becomes

$\int_{0}^{1}u^{99}du$ . (Don't forget to change the bounds of integration by plugging them into our equation for )

$= [\frac{u^{100}}{100}]_{0}^{1}$

$=\frac{1}{100}$

Report an Error

Example Question #25 : Definite Integrals

Evaluate $\int_{1}^{e} \frac{\ln(x)}{x^2}dx$ .

Possible Answers:

e^2-8

$-\infty$

None of the other answers

$\frac{e-2}{e}$

Correct answer:

$\frac{e-2}{e}$

Explanation:

We proceed by using integration by parts.

$\int_{1}^{e} \frac{\ln(x)}{x^2}dx$ . Start

Let $u= \ln(x), dv=\frac{1}{x^2}dx$ , then we get $du = \frac{1}{x}dx, v=-\frac{1}{x}$
. Then using the integration by parts formula $uv - \int vdu$ , we get

$=[-\frac{\ln(x)}{x}]_{1}^{e} - \int_{1}^{e}-\frac{1}{x^2}dx$

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

$=-\frac{1}{e}+ \int_{1}^{e}x^{-2}dx$

$=-\frac{1}{e} +[-x^{-1}]^{e}_{1}$

$= -\frac{1}{e}+(-\frac{1}{e}+1)$

$=1-\frac{2}{e}$

$=\frac{e-2}{e}$

Report an Error

Example Question #26 : Definite Integrals

Evaluate the following integral:

$\int_{0}^{2\pi}7x\sin(x)dx$

Possible Answers:

$7\pi$

$-14\pi$

$-15\pi$

$14\pi$

Correct answer:

$-14\pi$

Explanation:

To evaluate this integral, we must integrate by parts, according to the following formula:

$\int u dv=uv-\int vdu$

So, we must assign our u and dv, and differentiate and integrate to find du and v, respectively:

u=7x, du=7dx

$dv=\sin(x)dx, v=-\cos(x)$

The derivative and integral were found using the following rules:

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

Note that we ignore the constant of integration.

Now, use the above formula:

$[-7(2\pi)\cos(2\pi)-7(0)\cos(0)]+7\int_{0}^{\2\pi} \cos(x)dx=-14\pi+[7\sin(2\pi)-7\sin(0)]$

Note that both the product of u and v and the integral are being evaluated from zero to $2\pi$ .

The integral was performed using the following rule:

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

Simplifying the above results, we get $[-14\pi-0]+[0-0]=-14\pi$ .

Report an Error

Example Question #21 : Definite Integrals

Find the area between y=sin(x) and y=sec^2(x) between $x:(0, \frac{\pi}{4})$

Possible Answers:

$\frac{\sqrt2}{2}+2$

$\frac{\sqrt2}{2}$

Correct answer:

$\frac{\sqrt2}{2}$

Explanation:

We can write this problem as:

$\int_{0}^{\frac{\pi}{4}}(sec^2(x)-sin(x))dx$

Integrating:

$\int(sec^2(x)-sin(x))dx=tan(x)+cos(x)+C=F(x)$

By the fundamental theorem of calculus:

$\int_{0}^{\frac{\pi}{4}}(sec^2(x)-sin(x))dx=F(\frac{\pi}{4})-F(0)$

$=[tan(\frac{\pi}{4})+cos(\frac{\pi}{4})]-[tan(0)+cos(0)]=[1+\frac{\sqrt2}{2}]-[1]$

$\int_{0}^{\frac{\pi}{4}}(sec^2(x)-sin(x))dx=\frac{\sqrt2}{2}$

Report an Error

Example Question #391 : Integrals

$Evaluate:\int_{0}^{5}(x^2+x)dx$

Possible Answers:

$\frac{325}{6}$

$\frac{175}{6}$

$\frac{200}{6}$

$\frac{325}{2}$

$\frac{250}{3}$

Correct answer:

$\frac{325}{6}$

Explanation:

Compute the Indefinite Integral

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

Evaluate the integral

$(\frac{5^3}{3}+\frac{5^2}{2})-(\frac{0^3}{3}+\frac{0^2}{2})$

$(\frac{125}{3}+\frac{25}{2})-(0+0)$

$\frac{250}{6}+\frac{75}{6}=\frac{325}{6}$

Report an Error

Example Question #29 : Definite Integrals

Suppose u=ax , where is a constant

Find such that $\int_{0}^{2} 5u dx =2$

Possible Answers:

u=x

u=5x

$u=\frac{1}{5}x$

$u=\frac{5}{2}x$

Correct answer:

$u=\frac{1}{5}x$

Explanation:

$\int 5ax dx= \frac{5}{2}ax^2 +C= F(x)$

By the fundamental theorem of calculus:

$\int_{0}^{2} 5u dx =F(2)-F(0)=2$

$\frac{5}{2}a*2^2-0=2$

10a=2

$a=\frac{1}{5}$

$u=\frac{1}{5}x$

Report an Error

Example Question #21 : Definite Integrals

Evaluate this integral.

$\int_{2}^{0} 3+4x dx$

Possible Answers:

Answer not listed

Correct answer:

Explanation:

$\int_{a}^{b} f(x) dx$

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

In this case, f(x) = 3 + 4x .

The antiderivative is F(x) = 3x + 2x^2 .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{2}^{0} 3 + 4x dx = \left (3x + 2x^2 \right ) _{2}^{0}$

$= 3(0) + 2(0)^2 - \left (3(2) + 2(2)^2 \right )$

= - 14

Report an Error

View Calculus 2 Tutors

Philipp
Certified Tutor

University of Nevada-Las Vegas, Bachelor of Science, Mechanical Engineering.

View Calculus 2 Tutors

Andrew
Certified Tutor

Louisiana State University Eunice, Bachelor of Science, Computer Software Engineering.

View Calculus 2 Tutors

Reuben
Certified Tutor

University of Alberta, Bachelor of Science, Mathematics.

All Calculus 2 Resources

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

Popular Subjects

SAT Tutors in Houston, Spanish Tutors in Washington DC, English Tutors in Boston, GMAT Tutors in Denver, ACT Tutors in Dallas Fort Worth, ACT Tutors in New York City, Reading Tutors in Houston, Physics Tutors in Miami, MCAT Tutors in San Francisco-Bay Area, English Tutors in Houston

Popular Courses & Classes

SAT Courses & Classes in San Diego, SSAT Courses & Classes in Miami, SSAT Courses & Classes in Atlanta, GRE Courses & Classes in Chicago, Spanish Courses & Classes in Boston, MCAT Courses & Classes in Los Angeles, Spanish Courses & Classes in Philadelphia, Spanish Courses & Classes in Chicago, MCAT Courses & Classes in Atlanta, SAT Courses & Classes in Chicago

Popular Test Prep

LSAT Test Prep in Atlanta, SSAT Test Prep in Phoenix, GRE Test Prep in Chicago, LSAT Test Prep in Washington DC, MCAT Test Prep in Phoenix, LSAT Test Prep in San Diego, ACT Test Prep in Washington DC, SAT Test Prep in Denver, MCAT Test Prep in Denver, SSAT 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 : Definite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #31 : Finding Integrals

Example Question #22 : Definite Integrals

Example Question #31 : Finding Integrals

Example Question #21 : Definite Integrals

Example Question #25 : Definite Integrals

Example Question #26 : Definite Integrals

Example Question #21 : Definite Integrals

Example Question #391 : Integrals

Example Question #29 : Definite Integrals

Example Question #21 : Definite Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: