Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Integral Expressions

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 #111 : How To Find Integral Expressions

Evaluate the integral:

$\int \frac{dx}{\sqrt{x+5}}$

Possible Answers:

$\ln(x+5)^{\frac{1}{2}}+C$

$\ln(\sqrt{x+5})+C$

$(x+5)^{\frac{1}{2}}+C$

$2\ln(\sqrt{x+5})+C$

Correct answer:

$2\ln(\sqrt{x+5})+C$

Explanation:

To integrate, we must make the following subsitution:

$u=(x+5)^{\frac{1}{2}}, du=\frac{1}{2}(x+5)^{-\frac{1}{2}}dx$

Now, rewrite the integral, and integrate:

$2\int \frac{du}{u}=2\ln \left | u\right |+C$

The following rule was used for the integration:

$\int \frac{dx}{x}=\ln \left | x\right |+C$

Finally, replace u with our original term:

$2\ln(\sqrt{x+5})+C$

Note that the absolute value sign went away, because the square root is always positive.

Report an Error

Example Question #111 : Equations

Evaluate the following integral:

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

Possible Answers:

$e^x+\ln(x^2+3)+C$

$e^x+\sin(x^2+3)+C$

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

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

Correct answer:

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

Explanation:

To integrate, we must split the integral in two:

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

The first integral is:

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

and is identical to the rule used to integrate.

The second integral is performed using the following substitution:

u=x^2+3, du=2xdx

Now, rewrite the integral and integrate:

$\frac{1}{2}\int \cos(u)du=\frac{1}{2}\sin(u)+C$

We used the following rule to integrate:

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

Finally, replace u with the original term:

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

Report an Error

Example Question #112 : How To Find Integral Expressions

Evaluate the following integral:

$\int (15\sec(x)+x^2)dx$

Possible Answers:

$\ln\left | \sec(x)+\tan(x)\right |+\frac{x^3}{3}+C$

$15\ln\left | \sec(x)+\tan(x)\right |+\frac{x^3}{3}+C$

$15\tan(x)+\frac{x^3}{3}+C$

$15\ln\left | \sec(x)+\tan(x)\right |+x^3+C$

Correct answer:

$15\ln\left | \sec(x)+\tan(x)\right |+\frac{x^3}{3}+C$

Explanation:

The integral is equal to
$15\ln\left | \sec(x)+\tan(x)\right |+\frac{x^3}{3}+C$

and was found using the following rules:

$\int \sec(x)=\ln\left | \sec(x)+\tan(x)\right |+C$ ,

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

Report an Error

Example Question #111 : Writing Equations

Evaluate the following integral:

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

Possible Answers:

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

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

$x^3-2\sin(x)+C$

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

Correct answer:

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

Explanation:

The integral is equal to

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

and we used the following rules for integration:

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

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

Report an Error

Example Question #1161 : Functions

Evaluate the integral:

$\int \left(8x^4+\sec^2(x)+\frac{1}{x^2+1}\right)dx$

Possible Answers:

$8x^5+\tan(x)+\frac{1}{2}\ln(x^2+1)+C$

$\frac{8}{5}x^5+\tan(x)+\arccos(x)+C$

$8x^5+\tan(x)+\arctan(x)+C$

$\frac{8}{5}x^5+\tan(x)+\arctan(x)+C$

Correct answer:

$\frac{8}{5}x^5+\tan(x)+\arctan(x)+C$

Explanation:

The integral was performed using the following rules:

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

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

$\int \frac{dx}{x^2+1}=\arctan(x)+C$

Applying the above rules we get the following.

$\frac{8}{5}x^5+\tan(x)+\arctan(x)+C$

Report an Error

Example Question #113 : How To Find Integral Expressions

Evaluate the following integral:

$\int xe^{3x^2}dx$

Possible Answers:

$\frac{1}{6}e^{3x^2}+C$

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

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

$\frac{1}{6}e^{2x^2}+C$

Correct answer:

$\frac{1}{6}e^{3x^2}+C$

Explanation:

To integrate, we must make the following substitution:

u=3x^2, du=6xdx

Now, rewrite the integral and integrate:

$\frac{1}{6}\int e^udu=\frac{1}{6}e^u+C$

The following rule was used to integrate:

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

Finally, replace u wiht our original term:

$\frac{1}{6}e^{3x^2}+C$

Report an Error

Example Question #114 : How To Find Integral Expressions

$\int (e^x+x\cos(x^2))dx$

Possible Answers:

$e^x-\cos(x^2)+C$

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

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

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

Correct answer:

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

Explanation:

The integral can be split into two seperate ones:

$\int e^xdx+\int x\cos(x^2)dx$

The first integral is simple:

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

and is identical to the rule used.

The second integral can be performed using the following substitution:

u=x^2, du=2xdx

Rewrite and integrate:

$\frac{1}{2} \int \cos(u)du=\frac{1}{2}\sin(u)+C$

The following rule was used to integrate:

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

Finally, replace u with the original term and add the integrals together:

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

Report an Error

Example Question #118 : How To Find Integral Expressions

Given that the piecewise function f(x)=cos(x) for $x\leq0$ and $f(x)=e^{2x}$ for x>0 , find $\int_{-\frac{\pi }{2}}^{2}f(x)dx$ .

Possible Answers:

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

$\frac{1-e^4}{2}$

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

$\frac{1+e^4}{4}$

$\frac{1+e^4}{2}$

Correct answer:

$\frac{1+e^4}{2}$

Explanation:

Since we have a piecewise function we will need to set up an integral with two parts. One will f(x)=cos(x) from $-\frac{\pi}{2}$ to zero and the other, $f(x)=e^{2x}$ from zero to two.

Setting up the integral and plugging in the bounds looks like,

$\\ \int_{-\frac{\pi }{2}}^{2}f(x)dx=\int_{-\frac{\pi }{2}}^{0}f(x)dx+\int_{0}^{2}f(x)dx\\ \\=\int_{-\frac{\pi }{2}}^{0}cos(x)dx+\int_{0}^{2}e^{2x}dx\\ \\=sin(x)]\begin{matrix}-\frac{\pi }{2}, \end{matrix}0+\frac{e^{2x}}{2}]\begin{matrix}0, \end{matrix}2\\ \\=[sin(0)-sin(-\frac{\pi }{2})]+[\frac{e^{4}}{2}-\frac{e^0}{2}]\\ \\=[0-(-1)]+[\frac{e^4}{2}-\frac{1}{2}]\\ \\=1-\frac{e^4-1}{2}\\ \\=\frac{2+e^4-1}{2}\\ \\=\frac{1+e^4}{2}$ .

Report an Error

Example Question #115 : How To Find Integral Expressions

Find the general anti-derivative of ${} x^2 e^x$ .

Possible Answers:

${} \frac{x^3e^x}{3}+C$

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

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

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

Correct answer:

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

Explanation:

We have a product of two easy functions, so to get our answer we need to use integration by parts. The general formula for this is:

$\int u dv = uv- \int v du$ .

Choose u=x^2 and dv=e^x dx . Then du=2xdx and v=e^x.

So integration by parts tells us that our antiderivative equals

${} x^2e^x-\int 2xe^x dx.$

Now we must apply integration by parts again, because we still have an integral of a product of two functions. Choose u=2x and dv=e^x dx again, so du=2dx and v=e^x .

Then we have

${} \int 2xe^x dx = 2xe^x-\int2e^x dx.$

But of course, 2e^x is its own antiderivative, because we can pull out constants and e^x is its own anti-derivative.

So combining these two results we get our final answer:

${} \int x^2e^x dx = x^2e^x-2xe^x+2e^x+C.$

Report an Error

Example Question #115 : How To Find Integral Expressions

Write the integral expression for position p(t) given acceleration a(t) .

Possible Answers:

$p(t)=\int \left[\int a(t)dt\right]dt$

$p(t)=\frac{\mathrm{d^2} }{\mathrm{d} t^2}\int \left[\int a(t)dt\right]dt$

$p(t)=\frac{\mathrm{d^2}a(t) }{\mathrm{d} t^2}$

$p(t)=\frac{\mathrm{d} }\left{\mathrm{d} t}[\int a(t)dt\right]$

Correct answer:

$p(t)=\int \left[\int a(t)dt\right]dt$

Explanation:

The indefinite integral of acceleration is velocity. The indefinite integral of velocity is position. We can write this as:

$v(t)=\int a(t)dt$ , where a(t) is acceleration and v(t) is velocity

$p(t)=\int v(t)dt$ , where p(t) is position

$p(t)=\int \left[\int a(t)dt\right]dt$

Report an Error

View Calculus Tutors

Shakeel
Certified Tutor

View Calculus Tutors

Morgan
Certified Tutor

Catholic University of America, Bachelor of Science, International Finance.

View Calculus Tutors

John
Certified Tutor

The University of Texas at Arlington, Bachelor of Science, Mathematics. Stanford University, Master of Science, Aerospace Eng...

All Calculus 1 Resources

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

Popular Subjects

Physics Tutors in San Francisco-Bay Area, ISEE Tutors in Chicago, SSAT Tutors in San Diego, French Tutors in San Diego, Reading Tutors in Boston, GRE Tutors in San Diego, English Tutors in Miami, ISEE Tutors in Houston, Reading Tutors in New York City, Calculus Tutors in Houston

Popular Courses & Classes

Spanish Courses & Classes in Miami, Spanish Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Phoenix, GRE Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Los Angeles, Spanish Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Phoenix, Spanish Courses & Classes in New York City, SAT Courses & Classes in Boston, ACT Courses & Classes in Chicago

Popular Test Prep

LSAT Test Prep in Philadelphia, ISEE Test Prep in Seattle, ISEE Test Prep in Dallas Fort Worth, GMAT Test Prep in Philadelphia, LSAT Test Prep in Chicago, GRE Test Prep in San Francisco-Bay Area, GRE Test Prep in Denver, SAT Test Prep in Chicago, LSAT Test Prep in Houston, GRE Test Prep in Phoenix

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 : Integral Expressions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #111 : How To Find Integral Expressions

Example Question #111 : Equations

Example Question #112 : How To Find Integral Expressions

Example Question #111 : Writing Equations

Example Question #1161 : Functions

Example Question #113 : How To Find Integral Expressions

Example Question #114 : How To Find Integral Expressions

Example Question #118 : How To Find Integral Expressions

Example Question #115 : How To Find Integral Expressions

Example Question #115 : How To Find Integral Expressions

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: