Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Writing Equations

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #71 : Integral Expressions

Find the indefinite integral.

$\int4x^2-e^{3x}+\sec (x)^2 dx$

Possible Answers:

$\frac{4}{3}x^3-\frac{e^{3x}}{3}+\tan (x)+C$

$x^3-\frac{e^{3x}}{3}+\tan (x)+C$

None of these

$\frac{2}{3}x^3-\frac{e^{3x}}{3}+\tan (x)+C$

$\frac{4}{3}x^3-e^{3x}+\tan (x)+C$

Correct answer:

$\frac{4}{3}x^3-\frac{e^{3x}}{3}+\tan (x)+C$

Explanation:

For this integral you need to know an integration rule for each part of the problem.

$\int x^ndx=\frac{x^{n+1}}{n+1}$

Using this rule we can get the integral of the first part

$\int4x^2dx=\frac{4}{3}x^3$ .

For the second part we must know that

$\int e^xdx=e^x$

$\int-e^{3x}dx=-\frac{e^{3x}}{3}$ .

and finally the last part follows the rule exactly

$\int \sec (x)^2dx=\tan (x)$ .

We must also include a C for the constant of integration as the integral is indefinite.

Thus the final answer is

$\frac{4}{3}x^3-\frac{e^{3x}}{3}+\tan (x)+C$ .

Report an Error

Example Question #2151 : Calculus

Give the integral expression for the velocity of an object falling from initial velocity V(0)=G towards the earth with acceleration $-32\frac{ft}{s^2}$ , between and seconds after it was released.

Possible Answers:

$V(t)=\int_{0}^{4}(-32+G)dt$

$V(t)=[\int_{0}^{4}(G)dt] -32$

$V(t)=[\int_{4}^{0}(-32)dt] +G$

$V(t)=[\int_{0}^{4}(-32)dt] +G$

Correct answer:

$V(t)=[\int_{0}^{4}(-32)dt] +G$

Explanation:

We know that the definite integral of acceleration between 0 and 4 seconds is the velocity. We know that the acceleration due to gravity is $-32\frac{ft}{s^2}$ . We also know that the inititial velocity is . This means that when t=0, V(t)=G . Therefore, the velocity function can be written as

$V(t)=[\int_{0}^{4}(-32)dt] +G$

Report an Error

Example Question #71 : Writing Equations

Evaluate the following integral:

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

Possible Answers:

$e^{2\cos(x)}+C$

$e^{\2sin(x)}+C$

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

$-\frac{1}{2}e^{2\cos(x)}+C$

Correct answer:

$-\frac{1}{2}e^{2\cos(x)}+C$

Explanation:

To integrate, we must make the following substitution:

$u=2\cos(x), du=-2\sin(x)dx$

We used the following rule to derivate:

$\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Now, rewrite the integral and integrate:

$-\frac{1}{2}\int e^u du=-\frac{1}{2}e^u+C$

We used the following rule for the integration:

$\int e^xdx=e^xdx+C$

Now, replace u with the original x-containing term to finish:

$-\frac{1}{2}e^{\2cos(x)}+C$

Report an Error

Example Question #71 : Integral Expressions

Evaluate the following integral:

$\int x^3\cos(x^4)dx$

Possible Answers:

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

$\frac{1}{4}\sin(x^4)$

$\sin(x^4)+C$

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

Correct answer:

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

Explanation:

To integrate, we must make the following subsitution:

u=x^4, du=4x^3 dx

We used the following rule to derivate:

$\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$

Now, after rearranging, we get the following integral:

$\frac{1}{4}\int \cos(u) du$

and after integrating we get

$\frac{1}{4}\sin(u)+C$

We used the following rule to integrate:

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

To finish, we plug our x term back in place of u:

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

Report an Error

Example Question #72 : Writing Equations

Evaluate the following integral:

$\int (x^4+2x^2+\sec^2(x))dx$

Possible Answers:

$x^5+\frac{2x^3}{3}+\tan(x)+C$

$\frac{x^5}{5}+\frac{2x^3}{3}-\tan(x)+C$

$\frac{x^5}{5}+\frac{x^3}{3}+\tan(x)+C$

$\frac{x^5}{5}+\frac{2x^3}{3}+\tan(x)+C$

Correct answer:

$\frac{x^5}{5}+\frac{2x^3}{3}+\tan(x)+C$

Explanation:

After integrating, we get

$\frac{x^5}{5}+\frac{2x^3}{3}+\tan(x)+C$

The integration was performed using the following rules:

$\int \sec^2(x)=\tan(x)+C$ , $\int x^ndx=\frac{1}{n+1}x^{n+1}+C$

Report an Error

Example Question #2152 : Calculus

Evaluate the following integral:

$\int (4x^2\sin(x^3)+e^x)dx$

Possible Answers:

$\frac{4}{3}\cos(x^3)+e^x+C$

$-\frac{4}{3}\cos(x^3)+e^x+C$

$-\frac{4}{3}\cos(x^3)+C$

$\frac{4}{3}\sin(x^3)+e^x+C$

Correct answer:

$-\frac{4}{3}\cos(x^3)+e^x+C$

Explanation:

In order to evaluate this integral, we must split the integrand into two seperate integrals:

$\int4x^2\sin(x^3)dx+\int e^xdx$

For the first integral, we need to make the following substitution:

u=x^3, du=3x^2dx

We found the derivative using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$

Now, rewrite the integral in terms of u and solve:

$\frac{4}{3}\int \sin(u)du=-\frac{4}{3}\cos(u)+C$

We integrated using the following rule:

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

For the second integral, we just integrate:

$\int e^xdx=e^x+C$

and we use the rule

$\int e^xdx=e^x+C$

Now, just add together the results of the two integrals (the integral of the sum is equal to the sum of the integrals):

$-\frac{4}{3}\cos(x^3)+e^x+C$

Report an Error

Example Question #73 : Writing Equations

Evaluate the following integral:

$\int \frac{1}{\sqrt{x+4}}e^{\sqrt{x+4}}dx$

Possible Answers:

$\ln(\sqrt{x+4})+C$

$2e^{\sqrt{x+4}}+C$

$-2e^{\sqrt{x+4}}+C$

$\frac{(x+4)^{-\frac{1}{2}}}{e^{x+4}}+C$

Correct answer:

$2e^{\sqrt{x+4}}+C$

Explanation:

To integrate, we must make the following substitution:

$u=\sqrt{x+4}, du=\frac{1}{2}(x+4)^{-\frac{1}{2}}dx$

We found the derivative using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$

Now, rewrite the integral in terms of u, and solve:

$2\int e^udu=2e^{u}+C$

We used the following integration rule:

$\int e^xdx=e^xdx+C$

Now, replace u with the original term containing x:

$2e^{\sqrt{x+4}}+C$

Report an Error

Example Question #71 : Writing Equations

Evaluate the following integral:

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

Possible Answers:

$\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^2}{2}+C$

$x^3+\frac{x^4}{4}+x^2+C$

$\frac{x^3}{3}+\frac{x^{4}}{4}+x^2+C$

$\frac{3x^3}{3}+\frac{x^4}{4}+x^2+C$

Correct answer:

$\frac{x^3}{3}+\frac{x^{4}}{4}+x^2+C$

Explanation:

When we integrate, we get

$\frac{x^3}{3}+\frac{x^{4}}{4}+x^2+C$

which was found using the rule

$\int x^ndx=\frac{1}{n+1}x^{n+1}+C$

Report an Error

Example Question #2155 : Calculus

Evaluate the following indefinite integral:

$\int(8x^{3}+3x^{2}+2x)dx$

Possible Answers:

$2x^{4}+3x^{3}+2x+C$

$2x^{5}+x^{4}+3x^{2}+C$

$2x^{4}+x^{3}+x^{2}+C$

$x^{4}+x^{3}+x^{2}+C$

Correct answer:

$2x^{4}+x^{3}+x^{2}+C$

Explanation:

To evaluate the integral the integral, use the inverse power rule:

$\int x^{n}dx=\frac{x^{n+1}}{n+1} n\neq1$

Applying that rule to this problem gives us the following for the first term:

$\int 8x^{3}=\frac{8x^{3+1}}{3+1}=\frac{8x^{4}}{4}=2x^{4}$

The following for the second term:

$\int 3x^{2}=\frac{3x^{2+1}}{2+1}=\frac{3x^{3}}{3}=x^{3}$

And the following for the third term:

$\int 2x=\frac{2x^{1+1}}{1+1}=\frac{2x^{2}}{2}=x^{2}$

We can combine these terms and add our "C" to get the final answer:

$2x^{4}+x^{3}+x^{2}+C$

Report an Error

Example Question #72 : Writing Equations

Evaluate the indefinite integral:

$\int 2cos(x)dx$

Possible Answers:

cos(2x)+C

2sin(x)+C

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

sin(2x)+C

Correct answer:

2sin(x)+C

Explanation:

To evaluate this integral, we have to remember the trig laws. When taking the integral of cosine, our answer is always sine. In this case we do the following:

Move the 2 outside of the integral:

$2\int cos(x)dx$

Evaluate the integral of cosine and add our "C"

2sin(x)+C

Report an Error

View Calculus Tutors

Andrew
Certified Tutor

University of Wisconsin-Stevens Point, Bachelor of Science, Mathematics. University of Wisconsin-Oshkosh, Master of Science, ...

View Calculus Tutors

Travis
Certified Tutor

Delaware State University, Bachelor of Science, Mathematics.

View Calculus Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Math Tutors in Washington DC, SSAT Tutors in Denver, Statistics Tutors in Houston, Computer Science Tutors in Los Angeles, MCAT Tutors in Miami, Physics Tutors in Los Angeles, French Tutors in San Francisco-Bay Area, GRE Tutors in San Francisco-Bay Area, ACT Tutors in Dallas Fort Worth, French Tutors in Denver

Popular Courses & Classes

SAT Courses & Classes in Seattle, ACT Courses & Classes in Boston, LSAT Courses & Classes in Chicago, MCAT Courses & Classes in Denver, GMAT Courses & Classes in Washington DC, SAT Courses & Classes in Atlanta, Spanish Courses & Classes in Los Angeles, ACT Courses & Classes in Seattle, MCAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Boston

Popular Test Prep

LSAT Test Prep in Atlanta, MCAT Test Prep in Philadelphia, MCAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Boston, LSAT Test Prep in San Diego, SSAT Test Prep in Atlanta, MCAT Test Prep in Los Angeles, LSAT Test Prep in Washington DC, ISEE Test Prep in Atlanta, ACT Test Prep in Boston

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 1 : Writing Equations

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #71 : Integral Expressions

Example Question #2151 : Calculus

Example Question #71 : Writing Equations

Example Question #71 : Integral Expressions

Example Question #72 : Writing Equations

Example Question #2152 : Calculus

Example Question #73 : Writing Equations

Example Question #71 : Writing Equations

Example Question #2155 : Calculus

Example Question #72 : Writing Equations

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: