Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Indefinite 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 #499 : Finding Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{(76cos(x + 6))}{3}-\frac{ 25}{(102x)})dx\end{align*}$

Possible Answers:

$912sin(x + 6) -\frac{ (15625ln(x))}{1061208}$

$\frac{(76sin(x + 6))}{3}-\frac{ (25ln(x))}{102}$

$\frac{(76sin(x + 6))}{3}-\frac{ (25ln(x))}{102}+C$

$912sin(x + 6) -\frac{ (15625ln(x))}{1061208}+C$

Correct answer:

$\frac{(76sin(x + 6))}{3}-\frac{ (25ln(x))}{102}+C$

Explanation:

$\begin{align*}\\&\int[cos(ax)]=\frac{sin(ax)}{a}\\&\int\frac{a}{u}=aln(u)=ln(u^{a})\\&\text{In deriving the inside of}\frac{(76cos(x + 6))}{3}\text{, the lone constant goes to zero.}\\&\text{The }a\text{ value for }\frac{25}{(102x)}\text{ is }\frac{25}{102}\\&\int(\frac{(76cos(x + 6))}{3}-\frac{ 25}{(102x)})dx=\frac{(76sin(x + 6))}{3}-\frac{ (25ln(x))}{102}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

Report an Error

Example Question #221 : Indefinite Integrals

Find the solution to the Integral:

$\int \frac{ln^7(x)}{x}dx$

Possible Answers:

8ln(x)+c

7ln^6(x)+c

$\frac{}1{}8ln^2(x)+c$

8ln^8(x)+c

$\frac{ln^8(x)}{8}+c$

Correct answer:

$\frac{ln^8(x)}{8}+c$

Explanation:

To Evaluate this Integral we must use substitution:

Let

u=ln(x) and $du=\frac{1}{x}dx$

Therefore:

$\int \frac{ln^7(x)}{x}dx=\int u^7du$

$\int u^7du=\frac{1}{8}u^8+c=\frac{1}{8}ln^8(x)+c$

Report an Error

Example Question #501 : Finding Integrals

Evaluate the integrand:

$\int \frac{dx}{9x+7}dx$

Possible Answers:

$\frac{1}{9}ln{x}+7x+c$

$ln(9x^{10}+7x)+c$

$\frac{1}{9}ln(9x+7)+c$

ln(9x+7)+c

ln(9x-7)+c

Correct answer:

$\frac{1}{9}ln(9x+7)+c$

Explanation:

You have to use substitution to solve the integrand:

Let u=9x+7 and du=9dx and

$dx=\frac{du}{9}$

Therefore:

$\int \frac{dx}{9x+7}=\frac{1}{9}\int \frac{du}{u}=\frac{1}{9}ln(9x+7)+c$

Report an Error

Example Question #2601 : Calculus Ii

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{(7sin(10x))}{4}+\frac{ (19x^{2})}{5})dx\end{align*}$

Possible Answers:

$\frac{(76x^{3})}{15}-\frac{ (35cos(10x))}{2}$

$\frac{(19x^{3})}{15}-\frac{ (7cos(10x))}{40}+C$

$\frac{(76x^{3})}{15}-\frac{ (35cos(10x))}{2}+C$

$\frac{(19x^{3})}{15}-\frac{ (7cos(10x))}{40}$

Correct answer:

$\frac{(19x^{3})}{15}-\frac{ (7cos(10x))}{40}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int[sin(ax)]=-\frac{cos(ax)}{a}\\&\text{The }a\text{ value for }\frac{(19x^{2})}{5}\text{ is }2\\&\text{The }a\text{ value for }\frac{(7sin(10x))}{4}\text{ is }10\\&\int(\frac{(7sin(10x))}{4}+\frac{ (19x^{2})}{5})dx=\frac{(19x^{3})}{15}-\frac{ (7cos(10x))}{40}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

Report an Error

Example Question #222 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(3\cdot 3^{(\frac{x}{3})} + 11x^{15})dx\end{align*}$

Possible Answers:

$\frac{(9\cdot 3^{(\frac{x}{3})})}{ln(3)}+\frac{ (11x^{16})}{16}+C$

$\frac{3^{(\frac{x}{3})}}{ln(3)}+\frac{ (2475x^{16})}{16}$

$\frac{(9\cdot 3^{(\frac{x}{3})})}{ln(3)}+\frac{ (11x^{16})}{16}$

$\frac{3^{(\frac{x}{3})}}{ln(3)}+\frac{ (2475x^{16})}{16}+C$

Correct answer:

$\frac{(9\cdot 3^{(\frac{x}{3})})}{ln(3)}+\frac{ (11x^{16})}{16}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\text{The }a\text{ value for }11x^{15}\text{ is }15\\&\text{The }a\text{ value for }3\cdot 3^{(\frac{x}{3})}\text{ is }\frac{1}{3}\\&\int(3\cdot 3^{(\frac{x}{3})} + 11x^{15})dx=\frac{(9\cdot 3^{(\frac{x}{3})})}{ln(3)}+\frac{ (11x^{16})}{16}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

Report an Error

Example Question #223 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{(41sin(9x))}{10}+\frac{ (2\cdot 3^{(7x)})}{21})dx\end{align*}$

Possible Answers:

$\frac{(2\cdot 3^{(7x)})}{(147ln(3))}-\frac{ (41cos(9x))}{90}+C$

$\frac{(2\cdot 3^{(7x)})}{(3ln(3))}-\frac{ (369cos(9x))}{10}$

$\frac{(2\cdot 3^{(7x)})}{(147ln(3))}-\frac{ (41cos(9x))}{90}$

$\frac{(2\cdot 3^{(7x)})}{(3ln(3))}-\frac{ (369cos(9x))}{10}+C$

Correct answer:

$\frac{(2\cdot 3^{(7x)})}{(147ln(3))}-\frac{ (41cos(9x))}{90}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int[sin(ax)]=-\frac{cos(ax)}{a}\\&\text{The }a\text{ value for }\frac{(2\cdot 3^{(7x)})}{21}\text{ is }7\\&\text{The }a\text{ value for }\frac{(41sin(9x))}{10}\text{ is }9\\&\int(\frac{(41sin(9x))}{10}+\frac{ (2\cdot 3^{(7x)})}{21})dx=\frac{(2\cdot 3^{(7x)})}{(147ln(3))}-\frac{ (41cos(9x))}{90}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

Report an Error

Example Question #224 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{7}{(1170x)}+\frac{ (45x^{7})}{8})dx\end{align*}$

Possible Answers:

$\frac{(7ln(x))}{1170}+\frac{ (45x^{8})}{64}+C$

$\frac{(187ln(x))}{1170}+\frac{ (45x^{8})}{8}+C$

$\frac{(7ln(x))}{1170}+\frac{ (45x^{8})}{64}$

$\frac{(187ln(x))}{1170}+\frac{ (45x^{8})}{8}$

Correct answer:

$\frac{(7ln(x))}{1170}+\frac{ (45x^{8})}{64}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int\frac{a}{u}=aln(u)=ln(u^{a})\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\text{The }a\text{ value for }\frac{7}{(1170x)}\text{ is }\frac{7}{1170}\\&\text{The }a\text{ value for }\frac{(45x^{7})}{8}\text{ is }7\\&\int(\frac{7}{(1170x)}+\frac{ (45x^{7})}{8})dx=\frac{(7ln(x))}{1170}+\frac{ (45x^{8})}{64}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

Report an Error

Example Question #225 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{(3cos(x + 19))}{14}+\frac{ (38x^{6})}{3})dx\end{align*}$

Possible Answers:

$\frac{(3sin(x + 19))}{14}+\frac{ (38x^{7})}{21}$

$\frac{(3sin(x + 19))}{14}+\frac{ (38x^{7})}{21}+C$

$\frac{(1083sin(x + 19))}{14}+\frac{ (456x^{7})}{7}+C$

$\frac{(1083sin(x + 19))}{14}+\frac{ (456x^{7})}{7}$

Correct answer:

$\frac{(3sin(x + 19))}{14}+\frac{ (38x^{7})}{21}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int[cos(ax)]=\frac{sin(ax)}{a}\\&\text{The }a\text{ value for }\frac{(38x^{6})}{3}\text{ is }6\\&\text{In deriving the inside of}\frac{(3cos(x + 19))}{14}\text{, the lone constant goes to zero.}\\&\int(\frac{(3cos(x + 19))}{14}+\frac{ (38x^{6})}{3})dx=\frac{(3sin(x + 19))}{14}+\frac{ (38x^{7})}{21}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

Report an Error

Example Question #226 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{7}{(6x)}+\frac{ 4}{(9x^{6})})dx\end{align*}$

Possible Answers:

$\frac{(7ln(x))}{6}-\frac{ 4}{(45x^{5})}+C$

$\frac{(7ln(x))}{6}-\frac{ 4}{(45x^{5})}$

$\frac{(343ln(x))}{216}-\frac{ 16}{(5x^{5})}$

$\frac{(343ln(x))}{216}-\frac{ 16}{(5x^{5})}+C$

Correct answer:

$\frac{(7ln(x))}{6}-\frac{ 4}{(45x^{5})}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int\frac{a}{u}=aln(u)=ln(u^{a})\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\text{The }a\text{ value for }\frac{7}{(6x)}\text{ is }\frac{7}{6}\\&\text{The }a\text{ value for }\frac{4}{(9x^{6})}\text{ is }-6\\&\int(\frac{7}{(6x)}+\frac{ 4}{(9x^{6})})dx=\frac{(7ln(x))}{6}-\frac{ 4}{(45x^{5})}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

Report an Error

Example Question #227 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{(13x^{6})}{3}+\frac{ (9x^{7})}{49})dx\end{align*}$

Possible Answers:

$\frac{(156x^{7})}{7}+\frac{ (9x^{8})}{8}$

$\frac{(156x^{7})}{7}+\frac{ (9x^{8})}{8}+C$

$\frac{(x^{7}\cdot (27x + 728))}{1176}$

$\frac{(x^{7}\cdot (27x + 728))}{1176}+C$

Correct answer:

$\frac{(x^{7}\cdot (27x + 728))}{1176}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\text{The }a\text{ value for }\frac{(9x^{7})}{49}\text{ is }7\\&\text{The }a\text{ value for }\frac{(13x^{6})}{3}\text{ is }6\\&\int(\frac{(13x^{6})}{3}+\frac{ (9x^{7})}{49})dx=\frac{(x^{7}\cdot (27x + 728))}{1176}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

Report an Error

View Calculus 2 Tutors

Adam
Certified Tutor

Michigan State University, Bachelor of Science, Mathematics. Ohio State University-Main Campus, Doctor of Philosophy, Mathema...

View Calculus 2 Tutors

Steven
Certified Tutor

University of Waterloo, Bachelor of Science, Chemical Engineering. McMaster University, Master of Science, Chemical Engineeri...

View Calculus 2 Tutors

Reuben
Certified Tutor

University of Alberta, Bachelor of Science, Mathematics.

All Calculus 2 Resources

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

Popular Subjects

LSAT Tutors in Houston, SSAT Tutors in Chicago, Physics Tutors in Boston, SSAT Tutors in New York City, LSAT Tutors in Atlanta, Chemistry Tutors in San Diego, Statistics Tutors in Atlanta, SSAT Tutors in Washington DC, Reading Tutors in Boston, English Tutors in Atlanta

Popular Courses & Classes

ACT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Atlanta, ISEE Courses & Classes in Houston, GRE Courses & Classes in Denver, ISEE Courses & Classes in Los Angeles, MCAT Courses & Classes in Houston, GMAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Washington DC, GMAT Courses & Classes in Miami, Spanish Courses & Classes in San Francisco-Bay Area

Popular Test Prep

GRE Test Prep in San Francisco-Bay Area, GMAT Test Prep in Miami, SSAT Test Prep in Miami, SSAT Test Prep in Seattle, SSAT Test Prep in Phoenix, ACT Test Prep in New York City, SAT Test Prep in Atlanta, GRE Test Prep in Boston, GMAT Test Prep in San Francisco-Bay Area, SSAT Test Prep in Denver

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 : Indefinite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #499 : Finding Integrals

Example Question #221 : Indefinite Integrals

Example Question #501 : Finding Integrals

Example Question #2601 : Calculus Ii

Example Question #222 : Indefinite Integrals

Example Question #223 : Indefinite Integrals

Example Question #224 : Indefinite Integrals

Example Question #225 : Indefinite Integrals

Example Question #226 : Indefinite Integrals

Example Question #227 : Indefinite Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: