Comparison Test for Improper Integrals

In this video, I will show you how to use the comparison test to determine if an improper integral is convergent or divergent. An improper integral is convergent if the the result of solving the integral produces a constant number, and it is divergent if the result is infinity. But sometimes it is not possible to solve the improper integral by hand, like we did in the previous video. Instead, you must use the comparison test theorem for improper integrals which takes a shortcut in four steps and the comparison test tells you if the improper integral is convergent or divergent. This is a concept which is important in Calculus 2.
I like to use the FEP-NL-SL-C acronym to remember which function is greater than which one. That means as x approaches infinity,
Factorial greater than
Exponential greater than
Polynomial (of any degree) greater than
XlogX greater than
Linear greater than
Square root (or x to the power of a constant between 0 and 1) greater than
Logarithms greater than
Constant.
I know that seems like a lot, but it is so useful to me on many exams. Let me know if this makes sense and please buy me coffee by subscribing :) It means a lot
SOLUTIONS TO THE PROBLEMS:
drive.google.c...
In the next video, I will show you how to find the arc length of a function.
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[@KHOATRANANH-po5pv]

anh giảng hay quá, anh nói tiếng anh dễ nghe mà chuẩn. Mong anh sản xuất thêm nhiều video hay và chất lượng nữa ạ.

[@QuocDatPhung]

Cám ơn em! Em nhớ chia sẽ cho các bạn học và đăng kí ủng hộ anh nhé. Những bài dạy toán của anh ở đây: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@yonghoon8805]

Thank you! Your videos are very helpful and you explain all that complicated math with a smile on your face, amazing!

[@QuocDatPhung]

Thank you very much Yonghoon! I'm glad you enjoy it! You can find all of my Calculus 2 videos in this playlist: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@tailingoff]

Thanks for showing the counter example and also letting me know that hit and trial is the way we approach these tests. 😊

[@QuocDatPhung]

You're very welcome! If you like my Calculus videos, you can find the rest of them here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@嘉元张-y4f]

Best youtube teacher

[@QuocDatPhung]

Thank you! You can find all of my Calculus II videos in this playlist here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@Evelyn-oy8li]

Thank you so much, finally i could understand this topic after confusing during class. Thankyou 🥺🥺🥺

[@QuocDatPhung]

Thanks so much! Please kindly subscribe! Also, you can find the rest of my Calculus videos here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@sa_0515]

Greetings from the University of Florida. Have a Midterm in Calc 2 in 2 days 😅 Thank you a lot

[@QuocDatPhung]

Thank you Sa0515! I'm glad you found my videos helpful! Please kindly share with your classmates to help them and subscribe to support me (it means a lot). As well, you can find all of my Calculus II videos in this link: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@vivienshaanika]

You are a good tutor, this is amazing and continue doing your best 🙌🙌👏👏👏

[@QuocDatPhung]

Hi Vivien! Thank you so much! Please kindly subscribe and share with your classmates. You can find all of my Calculus II videos here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@ssamirs1881]

What happens if the numerator is not always positive?

[@QuocDatPhung]

You can factor out the -1 and move it outside the integral, since it is a constant.

[@ssamirs1881]

@@QuocDatPhung No, I mean for the second step, like for example a sin(x).

[@QuocDatPhung]

@@ssamirs1881 In the description of a video, I attached a Google Drive link that contains an example of cos(x) on the numerator (on the last page). Let me know if that helps.

[@QuocDatPhung]

@@ssamirs1881 Oh I meant the description of this video. Let me know if you can find it.

[@ibrahimtunahandag6081]

Bro bor bro bro bro bro!!! why we did not choose 1/x instead of 1/x^2?

[@ibrahimtunahandag6081]

Is it because of main func. is odd?

[@QuocDatPhung]

@@ibrahimtunahandag6081 Please let me know the timestamp of this problem, so I can help you with the specific example.

[@Earlx3n_Music]

Just trial and error,, both 1/x^2 and 1/x are acceptable 👌

[@rajpandya737]

finally makes sense

[@QuocDatPhung]

Thanks Raj! Please subscribe and share; it means a lot!

[@simonsays_0826]

I love you

[@QuocDatPhung]

Thank you! Please don't forget to share with your classmates and kindly subscribe ~ you can find all of my Calculus II videos in this link: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@potty1444]

so good

[@QuocDatPhung]

Thank you Potty! Please subscribe and share with classmates! It means a lot!

[@potty1444]

@@QuocDatPhung i did !!!

[@QuocDatPhung]

@@potty1444 Thank you so much!

[@md.ashrafulislamsami6791]

Great video! I have a question. Most of the time, I get stuck on step 3.....in step 3, you showed exponensial > linear ..In step 3 , how can we easily calculate which is greater/smaller than what.....is there any chart or video related to it..That would be great help..Thank you.

[@QuocDatPhung]

Thank you Ashraful! That is a very good question.
I like to use the FEP-NL-SL-C acronym to remember which function is greater than which one. That means as x approaches infinity, Factorial > Exponential > Polynomial (of any degree) > XlogX > Linear > Square root (or x to the power of a constant between 0 and 1) > Logarithms > Constant. I know that seems like a lot, but it is so useful to me on many exams. Let me know if this makes sense and please don't forget to subscribe :) It means a lot

[@na.1221]

Thanks teacher please keep it up

[@QuocDatPhung]

Thank you for your kind words Nardos!

[@na.1221]

No problem but can you work English grammatical for e.g relative clause and relative pronoun

[@QuocDatPhung]

@@na.1221 That's a good idea Nardos! I am planning to make videos on English grammar in the future too! Stay tuned!

[@na.1221]

Ok i will expect

[@QuocDatPhung]

@@na.1221 Awesome!

[@mertyilmaz2442]

Awesome as always :)

[@QuocDatPhung]

Thank you for your kind words Mert! :))))

[@NCont-i7g]

Truly appreciate those extra problems!

[@AndreaFerro-ks6bp]

No words, you are the best as always!!

[@QuocDatPhung]

Wow, thank you Andrea! If you really like my Calculus 2 videos, you can find the rest of them in this playlist: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@bossbabe6064]

Hi, thank you for this video! I have just one question, how do you determine the function g(x)? How do I know which function to use to compare to f(x)?

[@QuocDatPhung]

You can use this list (order from smallest to largest) to pick your g(x), but it does take a bit of practice to figure it out:
constant < log(x) < sqrt(x) < x < xlogx < polynomial < exponential < factorial

[@QuocDatPhung]

You can also see all of my Calculus II videos in this playlist: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@virendralakhawat1087]

❤❤❤❤❤

[@QuocDatPhung]

You're very welcome Virendralakhawat! Please don't forget to share with your classmates and kindly subscribe ~ you can find all of my Calculus II videos in this link: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@bhtkyav7420]

thank youuu everything is clear noww :)

[@QuocDatPhung]

That's great! You can find all of my Calculus II videos here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@nutritionbyelsa]

epic resource, thank you so much, so hard to find high quality resources on advanced math topics, sharing with my calc class at UBC, thank you : )

[@QuocDatPhung]

Thanks so much! I really appreciate it!!!

[@rainymc]

omg hi i saw your piazza post! thanks so much for sharing this vid!!!

[@QuocDatPhung]

@@rainymc You're very welcome! I don't have just this video, I have an entire playlist for Calculus II :) th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@ldhung308]

cảm ơn thầy vì bài giảng hữu ích

[@QuocDatPhung]

Cám ơn em! Em chia sẻ với các bạn khác học nhé.

[@THEoneandonlystika]

nice one

[@QuocDatPhung]

Thanks Messi! You can find all of my Calculus II videos in the following link (don't forget to share and kindly subscribe!): th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@gaming3rbbi393]

لك بتشرح احسن من العرب

[@QuocDatPhung]

Thank you! Please don't forget to share with your classmates and kindly subscribe ~ you can find all of my Calculus II videos in this link: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@Eagle._._.]

Thank you for saving my life

[@QuocDatPhung]

Thanks Eagle! If you enjoyed my clip, please do hit that subscribe button and share with classmates. As well, you can find the rest of my Calculus videos in this playlist: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@sahilbhagat7206]

Thank you very much to clearing concepts so easily.

[@QuocDatPhung]

Thanks so much Sahilbhagat! Please kindly subscribe and share with your classmates. You can find all of my Calculus II videos here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@bilgexo5973]

i hope you live happily forever. thank u for the video

[@QuocDatPhung]

Thank you and you too! Please don't forget to subscribe and share my calculus playlist with your classmates! th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@mrmansir3734]

very well made videos. I liked your improper integral video too

[@QuocDatPhung]

Thanks Mr.Mansir! Please don't forget to share with your classmates and kindly subscribe ~ you can find all of my Calculus II videos in this link: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@reemsayed9671]

You are so great may God bless you 💜

[@QuocDatPhung]

Thank you so much! You are awesome too! Please don't forget to subscribe and share my Calculus playlist here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@HanifaNasala]

Thank you, it's very easy to follow and understand😊

[@QuocDatPhung]

You're very welcome! Please don't forget to share with your classmates and kindly subscribe ~ you can find all of my Calculus II videos in this link: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@reemsayed9671]

Could you please include a translation into Arabic?!

[@QuocDatPhung]

Hi Farida! Thanks for letting me know! I will do some research on how to include other translations :)

[@reemsayed9671]

@@QuocDatPhung thanks for paying attention waiting for all your awesome coming vedios with translation into arabic🥰

[@QuocDatPhung]

@@reemsayed9671 Thank you! Please don't forget to subscribe and share with classmates :)) It means a lot!

[@andikusnadi1979]

cool

[@NCont-i7g]

Truly appreciate those extra problems!

[@QuocDatPhung]

You're very welcome! You can find all of my Calculus II videos here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@matthewwaite2679]

Clear and concise!

[@QuocDatPhung]

Thank you Matthew! Good luck on your tests!

[@voltaire8828]

We love you Dat, cant imagine life without you

[@QuocDatPhung]

You're welcome! I'm glad you like it! You can find the rest of my Calculus videos in this playlist: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@AungKhantPaing-q9q]

I am loving you so much!

[@QuocDatPhung]

Thanks a lot! Please kindly subscribe and share! Also, you can find all of my Calculus II videos here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@spaceclottey6250]

super clear, thank you

[@QuocDatPhung]

You're very welcome! You can find the rest of my Calculus 2 videos here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@richmondoseitwumasi9975]

Thank you very much 😊

[@QuocDatPhung]

You're so welcome Richmond! If you really like my Calculus 2 videos, you can find the rest of them in one place here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@stanleywainaina1413]

Thank you mate

[@QuocDatPhung]

You're very welcome Stanley! Here's the rest of my Calculus 2 videos in this playlist: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@kudakwashemcmillengomba603]

Watching this video a few minutes before my test😮....Will come and give some feedback after😂

[@QuocDatPhung]

Great! How was the test? You can find all of my Calculus II videos here: th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html

[@kudakwashemcmillengomba603]

This video was so helpful 😭...I just followed the steps nd I sailed through. Definitely subscribing here!!

[@ibrahimtunahandag6081]

Bro why did u do that! Why did you split it from 1?

[@QuocDatPhung]

Which problem are you referring to? Please give me the timestamp. I have to split it if there is a vertical asymptote at x=1

[@ibrahimtunahandag6081]

first one@@QuocDatPhung

[@ibrahimtunahandag6081]

like...are we always choosing 1 as a start point?@@QuocDatPhung

[@QuocDatPhung]

@@ibrahimtunahandag6081 I see. You mean the problem at 1:34. This integral goes from 0 to infinity. Now, if you look at the top right corner of the video, the first step is to make sure that the integral goes from 1 to infinity (that's just the way they want you to do it). Since the integral is from 0 to infinity, then we split it into two integrals, one going from 0 to 1, and the other going from 1 to infinity. Let me know if that makes sense.

[@ibrahimtunahandag6081]

@@QuocDatPhung so we basically choose 1. But why not 2, 3, 4 or e?

[@emberlynnguyen3458]

Your explanations are really comprehensible! Thank you so much!

[@QuocDatPhung]

Thanks so much Emberlyn! You can find the rest of my calculus videos in this playlist here (please kindly subscribe and share also!) :th-cam.com/play/PLeTO6OT3-FKmuCeO97iKt_Aibx-a938JA.html