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 #24 : Solving Integrals By Substitution

Evaluate the following indefinite integral using the substitution method.

$\int 3(x\sec(x^3))^2 dx$

Possible Answers:

$\tan(\frac{x^4}{4})+C$

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

$\tan(x^3) + C$

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

$3\sec^2(x^3)+C$

Correct answer:

$\tan(x^3) + C$

Explanation:

The integral can be expanded by distributing the exponent.

$\int3(x\sec(x^3))^2dx=\int3x^2\sec^2(x^3)dx$

We will make the following substitution:

u=x^3 .

Differentiating both sides yields

du=3x^2dx .

We can then substitute the left hand side of each equation into our integral and evaluate it now.

$\int3x^2\sec^2(x^3)dx=\int\sec^2(u)du=\tan(u)+C.$

Finally, we substitute the original variable back into the expression:

$\tan(u)+C=\tan(x^3)+C$ .

Report an Error

Example Question #28 : Solving Integrals By Substitution

Solve:

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

Possible Answers:

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{1}{8}$

Correct answer:

Explanation:

Use substitution:

u = sin(x) , du = cos(x)dx

Plug the and into the regular equation, but no need to worry about the bounds yet:

$\int udu$ $= \frac{1}{2}u^{2}$

Plug back into the integrated equation from above and evaluate from to $\pi$ .

$= \frac{1}{2} (sin(x))^2$

$\frac{1}{2} (sin(\pi)^2 - sin(0)^2) = 0$

Report an Error

Example Question #961 : Integrals

Solve:

$\int \frac{(ln(x))^2}{x}dx$

Possible Answers:

None of the chocies.

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

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

$\frac{1}{3} (ln(x))^3+C$

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

Correct answer:

$\frac{1}{3} (ln(x))^3+C$

Explanation:

Use substitution integration:

u = ln(x)

$du=\frac{1}{x}dx$

$\int u^2 du=\frac{1}{3}u^3+C=\frac{1}{3}(ln(x))^3+C$

Report an Error

Example Question #22 : Solving Integrals By Substitution

What is the integral of $xe^{x^2}$ ?

Possible Answers:

$\frac{e^{x^2}}{2x}$

$e^{x^2}$

$\frac{e^{x^2}}{x}$

$\frac{1}{2}e^{x^2}$

Correct answer:

$\frac{1}{2}e^{x^2}$

Explanation:

Use substitution:

u = x^2

du = 2xdx

$\frac{du}{2} = xdx$

$\int \frac{e^{u}}{2}du$ $= \frac{e^{u}}{2}$ $= \frac{e^{x^2}}{2}$

Substitute back in.

Report an Error

Example Question #961 : Integrals

$\int \frac{dx}{\sqrt{16-4x^2}}$

Possible Answers:

$\arcsin(\frac{x}{2})+C$

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

$\frac{1}{4}\arcsin(\frac{x}{2})+C$

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

$\frac{1}{2}\arcsin(\frac{x}{4})+C$

Correct answer:

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

Explanation:

To evaluate the integral, we must first perform the following substitution:

u=2x, du=2dx

Now, rewrite the integral and integrate:

$\frac{1}{2}\int \frac{du}{\sqrt{16-u^2}}=\frac{1}{2}\arcsin(\frac{u}{4})+C$

The integration was performed using the following rule:

$\int \frac{dx}{\sqrt{a^2-x^2}}=\arcsin(\frac{x}{a})+C$

Finally, replace u with the original term:

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

Report an Error

Example Question #32 : Solving Integrals By Substitution

Evaluate the following integral:

$\int \frac{dx}{\sqrt{x+14}}$

Possible Answers:

$\ln(x+14)^{\frac{1}{2}}+C$

$\frac{\sqrt{x+14}}{2}+C$

$2\sqrt{x+14}+C$

$\sqrt{x+14}+C$

Correct answer:

$2\sqrt{x+14}+C$

Explanation:

To evaluate the integral, we must make the following substitution:

$u=\sqrt{x+14}, du=\frac{1}{2}(x+14)^{-\frac{1}{2}}dx$

Now, rewrite the integral and integrate:

$\int2du=2u+C$

The integral was performed using the following rule:
$\int dx=x+C$

Finally, replace u with our original term:

$2\sqrt{x+14}+C$

Report an Error

Example Question #31 : Solving Integrals By Substitution

Evaluate the indefinite integral $\int 10e^{-x}dx$ .

Possible Answers:

$-5e^{-x}+C$

$-\frac{10e^{-x}}{x}+C$

None of the other answers

$10e^{-x}+C$

$-10e^{-x}+C$

Correct answer:

$-10e^{-x}+C$

Explanation:

We proceed as follows,

$\int 10e^{-x}dx$ . Start

$10\int e^{-x}dx$ . Factor out the 10.

Use u-substitution with u = -x , then taking derivates of both sides gives du = -1 dx .

$10\int -1e^u du$ . Substitute values

$-10\int e^u du$ . Factor out the negative.

-10e^u+C . The antiderivative of e^u is e^u . Don't forget .

$-10e^{-x}+C$ . Substitute back.

Report an Error

Example Question #31 : Solving Integrals By Substitution

Evaluate the indefinite integral $\int xe^{x^2} dx$ .

Possible Answers:

$2xe^{x^2}+C$

$\frac{e^{x^2}}{x^2} +C$

None of the other answers

$\frac{1}{2}e^{x^2}+C$

Not possible to integrate

Correct answer:

$\frac{1}{2}e^{x^2}+C$

Explanation:

We evaluate the integral as follows,

$\int xe^{x^2} dx$ . Start

Use u-substitution, let u=x^2 , then taking derivatives of both sides gives du = 2xdx . Divide both sides of this equation by , giving $\frac{1}{2}du = xdx$ . Now we can substitute out xdx, x^2 , and get

$\int e^{u} \frac{1}{2}du$

$\frac{1}{2}\int e^{u} du$ . Factor out the 1/2 .

$\frac{1}{2}e^u+C$ . Integrate e^u and add .

$\frac{1}{2}e^{x^2}+C$ . Substitute back

Report an Error

Example Question #31 : Solving Integrals By Substitution

Evaluate $\int^{15}_{8}x\sqrt{1+x}dx$ .

Possible Answers:

$\frac{246}{17}$

$\frac{4316}{15}$

None of the other answers.

$\frac{234}{521}$

Correct answer:

$\frac{4316}{15}$

Explanation:

We use u-substitution to evaluate this integral.

Let u = 1+x . Subtracting gives u-1=x , and taking derivatives gives du =dx (We subtract from both sides in order to make the expression under the square root as simple as possible). Then we have

$\int^{15}_{8}x\sqrt{1+x}dx$ . Start

$\int^{16}_{9}(u-1)\sqrt{u}du$ . Make our substitutions. (Make sure you change the bounds of integration too, by plugging and into u=1+x for ).

$\int^{16}_{9}(u-1)u^{\frac{1}{2}}du$ .

$\int^{16}_{9} u^{\frac{3}{2}}-u^{\frac{1}{2}}du$

$[\frac{2}{5}u^{\frac{5}{2}}-\frac{2}{3}u^{\frac{3}{2}}]^{16}_9$

$(\frac{2}{5}(16)^{\frac{5}{2}}-\frac{2}{3}(16)^{\frac{3}{2}})-(\frac{2}{5}(9)^{\frac{5}{2}}-\frac{2}{3}(9)^{\frac{3}{2}})$

$\frac{2}{5}(1024)-\frac{2}{3}(64)-\frac{2}{5}(243)+\frac{2}{3}(27)$

$\frac{2}{5}(781)-\frac{2}{3}(37)$

$\frac{1562}{5}-\frac{74}{3}$

$\frac{4316}{15}$

Report an Error

Example Question #36 : Solving Integrals By Substitution

Evaluate $\int^{\pi/2}_{0}\frac{2x\tan^{-1}(x^2)}{1+x^4}dx$

Possible Answers:

$\frac{\pi}{2}$

$\frac{\pi}{3}$

None of the other answers.

Correct answer:

None of the other answers.

Explanation:

The correct answer is $\frac{1}{2}(\tan^{-1}(\frac{\pi^2}{4}))^2$ .

We proceed as follows-

$\int^{\pi/2}_{0}\frac{2x\tan^{-1}(x^2)}{1+x^4}dx$ . Start

Evaluating this integral relies on the fact $\frac{d}{dx}\tan^{-1}(x) = \frac{1}{1+x^2}$ , and the Chain Rule for derivatives.

Use u-substitution $u = \tan^{-1}(x^2)$ , then we obtain $du = \frac{1}{1+(x^2)^2} \times 2x dx$

Our integral then becomes

$\int^{\tan^{-1}(\frac{\pi^2}{4})}_{0} u du$ after substitution. (The new upper bound on the integral cannot be simplified well, so we should leave it as is).

We then integrate to get

$[\frac{1}{2}u^2]^{\tan^{-1}(\frac{\pi^2}{4})}_0$

$\frac{1}{2}(\tan^{-1}(\frac{\pi^2}{4}))^2$

Report an Error

View Calculus 2 Tutors

Mohammed
Certified Tutor

University of Maryland-College Park, Bachelor of Science, Electrical Engineering. University of Maryland-College Park, Master...

View Calculus 2 Tutors

Pankaj
Certified Tutor

University of Delhi, Electrical Engineer, Electrical Engineering. Columbia University in the City of New York, Master of Arts...

View Calculus 2 Tutors

Robert
Certified Tutor

University of Guelph, Bachelor, Biochemistry.

All Calculus 2 Resources

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

Popular Subjects

SSAT Tutors in Los Angeles, English Tutors in Houston, Spanish Tutors in Seattle, Calculus Tutors in Miami, SAT Tutors in Houston, Math Tutors in New York City, GRE Tutors in Philadelphia, Math Tutors in Denver, Calculus Tutors in Washington DC, ISEE Tutors in Dallas Fort Worth

Popular Courses & Classes

SAT Courses & Classes in San Diego, ISEE Courses & Classes in Los Angeles, ACT Courses & Classes in Los Angeles, GMAT Courses & Classes in Seattle, GMAT Courses & Classes in Denver, ISEE Courses & Classes in New York City, SAT Courses & Classes in New York City, GRE Courses & Classes in Houston, GRE Courses & Classes in Phoenix, GMAT Courses & Classes in San Diego

Popular Test Prep

MCAT Test Prep in Los Angeles, GMAT Test Prep in Phoenix, GMAT Test Prep in Denver, ISEE Test Prep in Philadelphia, LSAT Test Prep in San Francisco-Bay Area, SAT Test Prep in Denver, ACT Test Prep in Atlanta, GRE Test Prep in Miami, GMAT Test Prep in Chicago, ACT 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 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #24 : Solving Integrals By Substitution

Example Question #28 : Solving Integrals By Substitution

Example Question #961 : Integrals

Example Question #22 : Solving Integrals By Substitution

Example Question #961 : Integrals

Example Question #32 : Solving Integrals By Substitution

Example Question #31 : Solving Integrals By Substitution

Example Question #31 : Solving Integrals By Substitution

Example Question #31 : Solving Integrals By Substitution

Example Question #36 : Solving Integrals By Substitution

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: