We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Solving Integrals by Substitution

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 #671 : Finding Integrals

Evaluate the integral with a substitution,

$\small \small \int\frac{ \sin(\ln x)}{x} dx$

Possible Answers:

$\small \small \small \cos(\ln x)+\frac{1}{x}+C$

$\small -\sin(\ln x)+\cos(\ln x)+C$

$\small \small - \cos(\ln x) + C$

$\small \frac{\ln x}{x}+\sin(\ln x)$

$\small \small -\sin(\ln x) + C$

Correct answer:

$\small \small - \cos(\ln x) + C$

Explanation:

$\small \small \int\frac{ \sin(\ln x)}{x} dx$

Let

$\small u = ln(x)$

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

$\small \small \small \small \int\frac{ \sin(\ln x)}{x} dx=\int\sin(u)du=-\cos(u)$

We can now convert this back to a function of $\small x$ by substituting $\small u = \ln x$ ,

$\small -\cos(u)=-\cos(\ln x)$

$\small \small \small \small \small \int\frac{ \sin(\ln x)}{x} dx = - \cos(\ln x)$

Report an Error

Example Question #91 : Solving Integrals By Substitution

Calculate the following integral: $\int \frac{x}{x+2}dx$

Possible Answers:

$x+2-ln\left |x+2 \right |+C$

$2x-2ln\left |x+2 \right |+C$

$x+2-\frac{1}{2}ln\left |x+2 \right |+C$

$x-2ln\left |x+2 \right |+C$

Correct answer:

$x-2ln\left |x+2 \right |+C$

Explanation:

$\int \frac{x}{x+2}dx$

Add 2 and subtract 2 from the numerator of the integrand: $\int \frac{x}{x+2}dx=\int \frac{x+2-2}{x+2}dx$ .

Simplify and apply the difference rule: $\int \frac{x+2-2}{x+2}dx=\int (\frac{x+2}{x+2}-\frac{x}{x+2})dx=\int dx-\int \frac{2}{x+2}dx$

Solve the first integral: $\int dx=x+C$ .

Make the following substitution to solve the second integral: u=x+2 du=dx

Apply the substitution to the integral: $\int \frac{2}{x+2}dx=2\int\frac{du}{u}$

Solve the integral: $2\int\frac{du}{u}=2ln\left | x+2\right |+C$

Combine the answers to the two integrals: $\int dx-\int \frac{2}{x+2}dx=x-2ln\left | x+2\right |+C$ .

Solution: $\int \frac{x}{x+2}dx=x-2ln\left | x+2\right |+C$

Report an Error

Example Question #672 : Finding Integrals

Evaluate the Integral:

$\int \frac{sin(x)}{2+cos(x)}dx$

Possible Answers:

ln(cos(x)+2)+c

ln(cos(x)-2)+c

ln(cos(x))+2+c

-ln(cos(x)-2)+c

-ln(2+cos(x))+c

Correct answer:

-ln(2+cos(x))+c

Explanation:

We use substitution to solve the problem:

Let u=2+cos(x) and du=-sin(x)dx $\Rightarrow sinxdx=-du$

Therefore:

$\int \frac{sin(x)}{2+cos(x)}dx=-\int \frac{1}{u}du=-ln(u)+c=-ln(2+cos(x))+c$

Report an Error

Example Question #1021 : Integrals

Evaluate

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

Possible Answers:

tan(x)+e^x +c

$e^{sinx}+c$

$e^{sinx} +cos(x) +c$

$e^{cosx}+c$

$e^{cosx}-sin(x)+c$

Correct answer:

$e^{sinx}+c$

Explanation:

Here we use substitution to solve for the integrand. Let u=sin(x) therefore du= cos(x)dx. Plug your values back in:

$\int cos(x)e^{sinx}dx = \int e^udu = e^u +c =e^{sinx}+c$

Report an Error

Example Question #92 : Solving Integrals By Substitution

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

Possible Answers:

0.3266

0.2466

0.3496

0.4581

1.3466

Correct answer:

0.4581

Explanation:

To integrate this expression, you have to use u substitution. First, assign your u expression:

u=x^2+1

du=2xdx

$\frac{1}{2}du=xdx$

Now, plug everything back in so you can integrate:

$\frac{1}{2}\int_{1}^{2}\frac{1}{u}du$

Now integrate:

$\frac{1}{2}\ln\left | u \right |$

From here substitute the original variable back into the expression.

$\frac{1}{2}\ln|x^2+1|$

Evaluate at 2 and then 1.

Subtract the results:

$\frac{1}{2}\left(\ln|2^2+1|-\ln|1+1| \right )$

$\frac{1}{2}\left(\ln|5|-\ln|2|\right)$

$\frac{\ln5-\ln2}{2}=0.4581$

Report an Error

Example Question #1022 : Integrals

Calculate the following integral:

$y=\int \sqrt{4-4x^{2}}dx$

Possible Answers:

$y=sin^{-1}x-sin(2sin^{-1}x)+C$

$y=sin^{-1}x+x\sqrt{1-x^2}+C$

$y=2sin^{-1}x-sin(2sin^{-1}x)+C$

$y=2sin^{-1}x-\frac{1}{2}sin(2sin^{-1}x)+C$

Correct answer:

$y=sin^{-1}x+x\sqrt{1-x^2}+C$

Explanation:

$y=\int \sqrt{4-4x^{2}}dx$

Factor out $\sqrt{4}$ from the integrand, and simplify:

$y=\int \sqrt{4}\sqrt{1-x^{2}}dx=\int 2\sqrt{1-x^{2}}dx=2\int \sqrt{1-x^{2}}dx$

Make the following substitution: $x=sin\theta$ $dx=cos\theta d\theta$

Plug the substitution into the integrand:

$y=2\int \sqrt{1-sin^{2}\theta }cos\theta d\theta$ .

Use the Pythagorean identity to make the following substitution, and simplify:

$\\y=2\int \sqrt{1-sin^{2}\theta }cos\theta d\theta\\=2\int \sqrt{cos^{2}\theta }cos\theta d\theta\\=2\int cos\theta cos\theta d\theta\\= 2\int cos^{2}\theta d\theta$

Apply the following identity to the integrand:

$cos^{2}\theta=\frac{1+cos(2 \theta )}{2}$ :

$y=2\int cos^{2}\theta=2\int \frac{1+cos(2) \theta }{2}d\theta$ .

Separate the integral into two separate integrals:

$y= 2\int \frac{1}{2}d\theta+2\int\frac{cos(2\theta )}{2}d\theta$ .

Solve the first integral:

$y=\theta+2\int\frac{cos(2\theta )}{2}d\theta$ .

Make the following substitution for the second integral:

$u=2\theta$ $du=2d\theta$ $\frac{1}{2}du=d\theta$ .

Apply the substitution, and solve the integral:

$2\int \frac{cos(2\theta )}{2}=2\int \frac{cosu}{2}\frac{1}{2}du=\frac{1}{2}sinu=\frac{1}{2}sin(2\theta )+C$ .

Combine answers for both integrals:

$y=2\int cos^{2}\theta=\theta+\frac{1}{2}sin(2\theta )+C$

Solve for $\theta$ :

$x=sin\theta$

$sin^{-1}x=\theta$

Plug values for $\theta$ back into solution to integral:

Recall that,

$sin(2\theta)=2sin(\theta)cos(\theta)$

and from above,

$\\x=sin(\theta) \\cos(\theta)=\sqrt{1-sin^2(\theta)}=\sqrt{1-x^2}$

Therefore,

$\\\theta+\frac{1}{2}sin(2\theta )+C \\=sin^{-1}+4sin(x)cos(x) \\=sin^{-1}(x)+x\sqrt{1-x^2}+C$ .

Report an Error

Example Question #681 : Finding Integrals

Evaluate the integral:

$\int xcos(\frac{1}{2}x^2)dx$

Possible Answers:

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

2sin(x^2)+C

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

2sin(x)+C

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

Correct answer:

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

Explanation:

You must use u substitution to solve this problem.

In this case

$u=\frac{1}{2}x^2$

and

du=xdx .

So the integral simplifies to

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

Report an Error

Example Question #2781 : Calculus Ii

Calculate the following integral:

$\int_{1}^{9}x\sqrt[3]{x-1}dx$

Possible Answers:

572

468/7

384/7

125/4

Correct answer:

468/7

Explanation:

Use $t=\sqrt[3]{x-1}$ as a substitute, then:

$t^3=x-1\:\Rightarrow\:x=t^3+1\:\Rightarrow\:dx=3t^2dt$ ;

Now, rewrite the boundaries of integration in terms of t:

$\newline t_1=\sqrt[3]{x_1-1}=\sqrt[3]{1-1}=0;\newline t_2=\sqrt[3]{x_2-1}=\sqrt[3]{9-1}=2;$

Rewrite the integral in terms of t:

$\\\int_{1}^{9}x\sqrt[3]{x-1}dx\\\\=\int_{0}^{2}(t^3+1)t\:3t^2\:dt\\\\=3\int_{0}^{2}t^6\:dt +3\int_{0}^{2}t^3\:dt\\\\=\frac{3*128}{7}+\frac{3*16}{4}\\\\=\frac{468}{7}$

Report an Error

Example Question #681 : Finding Integrals

Evaluate the following integral:

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

Possible Answers:

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

$e^{\frac{1}{x}}+c$

$\frac{1}{e^x}+c$

$-e^{\frac{1}{x}}+c$

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

Correct answer:

$-e^{\frac{1}{x}}+c$

Explanation:

We must use substitution to solve this integrand.

Let

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

and

$du=-\frac{1}{x^2}dx$

don't forget to divide buy -1 to isolate the right side

$-du=\frac{1}{x^2}dx$

putting all the values back in and pulling the negative sign out of the integral we get:

$\int \frac{e^{\frac{1}{x}}}{x^2}dx=-\int e^udu=-e^u+c=-e^{\frac{1}{x}}+c$

Report an Error

View Calculus 2 Tutors

Samantha
Certified Tutor

Middlesex County College, Associate in Science, Sustainability Studies.

View Calculus 2 Tutors

Amber
Certified Tutor

View Calculus 2 Tutors

Dossevi
Certified Tutor

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

All Calculus 2 Resources

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

Popular Subjects

ISEE Tutors in Houston, SSAT Tutors in Boston, Spanish Tutors in Atlanta, Chemistry Tutors in Denver, GMAT Tutors in San Francisco-Bay Area, GMAT Tutors in Seattle, Algebra Tutors in Miami, English Tutors in Seattle, Math Tutors in Denver, SAT Tutors in Chicago

Popular Courses & Classes

Spanish Courses & Classes in Philadelphia, LSAT Courses & Classes in Atlanta, ACT Courses & Classes in Dallas Fort Worth, ISEE Courses & Classes in Los Angeles, GMAT Courses & Classes in Denver, GMAT Courses & Classes in Chicago, SAT Courses & Classes in Boston, GMAT Courses & Classes in Atlanta, Spanish Courses & Classes in New York City, GRE Courses & Classes in San Francisco-Bay Area

Popular Test Prep

LSAT Test Prep in Washington DC, MCAT Test Prep in Atlanta, SAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Houston, SAT Test Prep in Seattle, SAT Test Prep in Miami, ISEE Test Prep in Denver, LSAT Test Prep in Houston, SAT Test Prep in Houston, SAT 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 2 : Solving Integrals by Substitution

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #671 : Finding Integrals

Example Question #91 : Solving Integrals By Substitution

Example Question #672 : Finding Integrals

Example Question #1021 : Integrals

Example Question #92 : Solving Integrals By Substitution

Example Question #1022 : Integrals

Example Question #681 : Finding Integrals

Example Question #2781 : Calculus Ii

Example Question #681 : Finding Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: