We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Functions

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 #1221 : Functions

Evaluate the following integral:

$\int 6x^2e^{4x^3+12}dx$

Possible Answers:

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

$\frac{1}{2(4x^3+2)}e^{4x^3+2}+C$

$\frac{1}{2}e^{4x^3+2}+C$

$\frac{1}{2}e^{4x^3+2}+2x+C$

Correct answer:

$\frac{1}{2}e^{4x^3+2}+C$

Explanation:

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

u=4x^3+2, du=12x^2dx

The derivative was found using the following rule:

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

Now, rewrite the integral and integrate:

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

The integral was performed using the following rule:

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

Finally, replace u with our original term:

$\frac{1}{2}e^{4x^3+2}+C$

Report an Error

Example Question #171 : Equations

Evaluate the following integral:

$\int( \csc(5x)+x)dx$

Possible Answers:

$\frac{\ln\left | \csc(5x)+\cot(5x)\right |}{5}+\frac{x^2}{2}+C$

$\frac{\ln\left | \csc(5x)-\cot(5x)\right |}{5}+{x^2}+C$

$\frac{\ln\left | \csc(5x)-\cot(5x)\right |}{5}+\frac{x^2}{2}+C$

${\ln\left | \csc(5x)-\cot(5x)\right |}+\frac{x^2}{2}+C$

Correct answer:

$\frac{\ln\left | \csc(5x)-\cot(5x)\right |}{5}+\frac{x^2}{2}+C$

Explanation:

To evaluate this integral, it is easier to split it into two integrals:

$\int \csc(5x)dx+ \int x dx$

For the first integral, the following substitution must be made:

u=5x, du=5dx

Rewriting the integral and integrating, we get:

$\frac{1}{5}\int \csc(u)du=\frac{1}{5}\ln\left | \csc(u)-\cot(u)\right |+C$

The following rule was used to integrate:

$\int \csc(x)dx=\ln\left | \csc(x)-\cot(x) \right |+C$

Replacing u with the original term, we get

$\frac{1}{5}\ln\left | \csc(5x)-\cot(5x)\right |+C$

Next, we evaluate the second integral:

$\int xdx=\frac{x^2}{2}+C$

which was found using the following rule:

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

Combining the results - and the constants of integration - we get

$\frac{\ln\left | \csc(5x)-\cot(5x)\right |}{5}+\frac{x^2}{2}+C$

Report an Error

Example Question #171 : Writing Equations

Set up, but don't evaluate an integral expression which will find the area of the region bound by the lines x=5 , x=25 , the x-axis, and the function v(x)

v(x)=14x^6+5x^4-3x^2+4

Possible Answers:

$\int_{25}^{25}14x^6+5x^4-3x^2dx$

$\int_{0}^{5}14x^6+5x^4-3x^2+4dx$

$\int_{0}^{20}14x^6+5x^4-3x^2+4dx$

$\int_{5}^{25}14x^6+5x^4-3x^2+4dx$

Correct answer:

$\int_{5}^{25}14x^6+5x^4-3x^2+4dx$

Explanation:

Set up, but don't evaluate an integral expression which will find the area of the region bound by the lines x=5 , x=25 , the x-axis, and the function v(x)

v(x)=14x^6+5x^4-3x^2+4

First, we need the correct limits of integration. We want 5 as the lower limit and 25 as the upper limit.

$\int_{5}^{25}$

Next, add in our function and the dx and we are all set!

$\int_{5}^{25}14x^6+5x^4-3x^2+4dx$

Report an Error

Example Question #171 : Integral Expressions

Set up, but do not evaluate, an integral expression for the area of the region bound by the y-axis, h(t), and the lines ,

h(t)=5e^t+18t^2-13

Possible Answers:

$\int_{0}^{6}5e^t+18t^2-13dt$

$\int_{0}^{5}5e^t+18t^2-13dt$

$\int_{5}^{6}5e^t+18t^2-13dt$

$\int_{5}^{6}5e^t+18t^2-13$

Correct answer:

$\int_{5}^{6}5e^t+18t^2-13dt$

Explanation:

Set up, but do not evaluate, an integral expression for the area of the region bound by the y-axis, h(t), and the lines ,

h(t)=5e^t+18t^2-13

To set up a definite integral, we first need our limits of integration:

In this case, they will be 5 and 6, because those are the lines which are the boundaries of our region.

$\int_{5}^{6}$

Finally, we want to include our function

$\int_{5}^{6}5e^t+18t^2-13dt$

Don't forget the dt, so that we know which variable we would integrate with respect to.

Report an Error

Example Question #171 : Equations

Evaluate the following integral:

$\int (t^2+10\csc(t)+e^{3t})dt$

Possible Answers:

$\frac{t^3}{3}+\ln\left | \csc(t)-\cot(t)\right |+\frac{e^{3t}}{3}+C$

$\frac{t^3}{3}+10\left | \csc(t)-\cot(t)\right |+\frac{e^{3t}}{3}+C$

$\frac{t^3}{3}+10\ln\left | \csc(t)+\cot(t)\right |+\frac{e^{3t}}{3}+C$

$\frac{t^3}{3}+10\ln\left | \csc(t)-\cot(t)\right |+\frac{e^{3t}}{3}+C$

$\frac{t^3}{3}+10\ln\left | \csc(t)-\cot(t)\right |+{e^{3t}}+C$

Correct answer:

$\frac{t^3}{3}+10\ln\left | \csc(t)-\cot(t)\right |+\frac{e^{3t}}{3}+C$

Explanation:

The integral is equal to

$\frac{t^3}{3}+10\ln\left | \csc(t)-\cot(t)\right |+\frac{e^{3t}}{3}+C$

and was found using the following rules:

$\int x^ndx=\frac{x^{n+1}}{n+1}+C$ , $\int \csc(x)dx=\ln\left | \csc(x)-\cot(x)\right |+C$ , $\int e^{ax}dx=\frac{e^{ax}}{a}+C$

Report an Error

Example Question #2255 : Calculus

Find the indefenite integral $\int \frac{sin(ln(x))}{x}dx$

Possible Answers:

$\frac{-cos(ln(x))}{x^2}$

cos(ln(x))+C

sin(ln(x))+cos(ln(x))+c

-cos(ln(x))+C

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

Correct answer:

-cos(ln(x))+C

Explanation:

We need to use substitution to solve. Let

u=ln(x)

Differentiating, gives

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

Solving for dx gives

dx=xdu

Plugging into the original integral gives

$\int \frac{sin(ln(x))}{x}dx=\int sin(ln(x))\cdot \frac{1}{x}dx=\int sin(u)\cdot \frac{1}{x}xdu$

Simplifying gives

$\int sin(u)du=-cos(u)+C$

Plugging back in for u gives

-cos(ln(x))+C

Report an Error

Example Question #171 : Writing Equations

Find the indefenite integral $\int x^2(x^3+3)^8dx$

Possible Answers:

$\frac{x^3(x^4+3)^9}{36}$

$\frac{x^3(x^4+3)^9}{108}$

$\frac{x^3(x^4+3)^8}{12}$

$\frac{(x^3+3)^9}{27}$

$\frac{x^3(x^3+3)^9}{81}$

Correct answer:

$\frac{(x^3+3)^9}{27}$

Explanation:

We need to use substitution to solve. Let

u=(x^3+3)^8

Differentiating gives

du=3x^2dx

Solving for dx, so that we can plug back in, gives

$dx=\frac{du}{3x^2}$

Plugging in gives

$\int x^2(x^3+3)^8dx=\int x^2 \cdot u^8 \cdot\frac{du}{3x^2}$

Simplifying gives

$\int u^8 \cdot\frac{du}{3}=\frac{1}{3}\int u^8du$

Integrating gives

$\frac{1}{3} \cdot \frac{u^9}{9}+C=\frac{u^9}{27}+C$

Plugging u back in gives

$\frac{(x^3+3)^9}{27}$

Report an Error

Example Question #2257 : Calculus

Find the indefenite integral $\int xsin(x)dx$

Possible Answers:

sin(x)-cos(x)+C

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

cos(x)-sin(x)+C

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

-xcos(x)+sin(x)+C

Correct answer:

-xcos(x)+sin(x)+C

Explanation:

We need to use integration by parts, which says

$\int udv=uv-\int vdu$

Let

u=x, dv=sin(x)dx

Differentiating u and integrating dv, gives

du=dx, v=-cos(x)

Plugging into the equation, gives

$\int xsin(x)dx=x \cdot (-cos(x))-\int (-cos(x))dx$

$=-xcos(x)+\int cos(x)dx$

Now, we can integrate the last term to get

=-xcos(x)+sin(x)+C

Report an Error

Example Question #172 : Integral Expressions

Find the indefenite integral $\int x^2ln(x)dx$

Possible Answers:

$y'=\frac{1}{x^2}+C$

$y'=\frac{x^3}{ln(x)}+C$

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

y'=x^2+C

$y'=\frac{x^2}{3}+C$

Correct answer:

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

Explanation:

We need to use integration by parts, which says

$\int udv=uv-\int vdu$

Let

u=ln(x), dv=x^2dx

Differentiating u and integrating dv, gives

$du=\frac{dx}{x}, v=\frac{x^3}{3}$

Plugging into the equation, gives

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

$=\frac{x^3}{3}ln(x)-\frac{1}{3}\int x^2dx$

Now, we can integrate the last term to get

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

$=\frac{x^3}{3}ln(x)-\frac{x^3}{9}+C$

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

Report an Error

Example Question #1224 : Functions

Solve the integral:

$\int e^{2s}\sec^2(e^{2s})ds$

Possible Answers:

$\tan(e^{2s})+C$

$\frac{1}{2}\ln\left |\sec(e^{2s})+\tan(e^{2s}) \right |+C$

$\frac{1}{2}\tan(e^{2s})+C$

$-\frac{1}{2}\tan(e^{2s})+C$

Correct answer:

$\frac{1}{2}\tan(e^{2s})+C$

Explanation:

To solve the integral, we must first make the following substitution:

$u=e^{2s}$

$du=2e^{2s}ds$

The derivative was found using the following rule:

$\frac{\mathrm{d} }{\mathrm{d} x} e^u = e^u \frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Now, rewrite the integral and integrate:

$\frac{1}{2}\int \sec^2(u)du=\frac{1}{2}\tan(u)+C$

The integral was found using the rule

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

Finally, replace u with the original term:

$\frac{1}{2}\tan(e^{2s})+C$ .

Report an Error

View Calculus Tutors

Rene Solo
Certified Tutor

University of the Philippines, Bachelor of Science, Mathematics. Prairie View A & M University, Master of Arts, Educational A...

View Calculus Tutors

Timothy
Certified Tutor

Bethune-Cookman University, Bachelor of Science, Mathematics. University of Phoenix, Master of Science, Computer and Informat...

View Calculus Tutors

Samuel
Certified Tutor

University of Ghana, Bachelor, Actuarial Science.

All Calculus 1 Resources

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

Popular Subjects

SSAT Tutors in Atlanta, MCAT Tutors in Seattle, Calculus Tutors in Phoenix, LSAT Tutors in San Diego, Math Tutors in Los Angeles, GMAT Tutors in San Francisco-Bay Area, SAT Tutors in Phoenix, GRE Tutors in New York City, Statistics Tutors in Philadelphia, ISEE Tutors in Chicago

Popular Courses & Classes

MCAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Philadelphia, LSAT Courses & Classes in Philadelphia, ACT Courses & Classes in Phoenix, GMAT Courses & Classes in Los Angeles, Spanish Courses & Classes in Seattle, ISEE Courses & Classes in Denver, Spanish Courses & Classes in Boston, Spanish Courses & Classes in San Diego, GRE Courses & Classes in Dallas Fort Worth

Popular Test Prep

ISEE Test Prep in Denver, MCAT Test Prep in Dallas Fort Worth, ACT Test Prep in Phoenix, ACT Test Prep in Denver, LSAT Test Prep in Washington DC, ISEE Test Prep in Miami, ISEE Test Prep in Dallas Fort Worth, MCAT Test Prep in Miami, SAT Test Prep in Dallas Fort Worth, SSAT Test Prep in Miami

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 : Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1221 : Functions

Example Question #171 : Equations

Example Question #171 : Writing Equations

Example Question #171 : Integral Expressions

Example Question #171 : Equations

Example Question #2255 : Calculus

Example Question #171 : Writing Equations

Example Question #2257 : Calculus

Example Question #172 : Integral Expressions

Example Question #1224 : Functions

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: