Integral of 2x/(x^2+a^2) (substitution)

[@IntegralsForYou]

𝐏𝐥𝐞𝐚𝐬𝐞 𝐒𝐔𝐁𝐒𝐂𝐑𝐈𝐁𝐄 𝐭𝐨 𝐦𝐲 𝐧𝐞𝐰 𝐜𝐡𝐚𝐧𝐧𝐞𝐥 𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞𝐬 𝐅𝐨𝐫𝐘𝐨𝐮! www.youtube.com/@DerivativesForYou 𝐓𝐡𝐚𝐧𝐤 𝐲𝐨𝐮! ❤

[@gabyshavit8539]

Hi I saw your video about the integral of square root of tanx just found it would be smarter to ask on your last video, I was wondering if maybe you can solve that integral like that: tanx=u^2 dx=2u/(u^4+1) Then 2*integral of u^2/(u^4+1)du can be written like: 2*integral of A/(u^2+i)du + 2*integral of B/(u^2-i)du Finding A and B are both equal to 1/2 We get integral of 1/(u^2+i)du + integral of 1/(u^2-i)du Let's start with the second integral: integral of 1/(u^2-i)du now we want to find square root of i for trig-sub after finding square root of i equals 1/4 + i/4 We substitute u=(1/4+i/4)sin(t) And get 1/(1/4+i/4)*integral of 1/cos(t)dt Now to solve the integral of 1/cos(t)dt I do not know how to do it I just know there is a special substitution for it can you please tell if what im doing is right? And also wanted to ask if you have videos for every kind of substitution method? Thank you so much for everything every time I watch one of your videos I learn more🫡

[@IntegralsForYou]

Hi! First of all, you can comment on any video, I will see it and answer there without any problem 😉 About the integral using complex numbers: yes, you can use complex numbers but at the end you should be able to simplify the expression and get a solution based on real numbers. I also think that you are doing it right but I am not used to use complex numbers for indefinite integration with real numbers. The integral of 1/cos(x) you can find it here kzbin.info/www/bejne/p364nqmcftSHqas and here kzbin.info/www/bejne/gmXSZneHh557n7M I don't think I have all kind of substitutions but here you have the substitutions I have: 1. u-substitution: Playlist on KZbin: kzbin.info/aero/PLpfQkODxXi4-7Nc5OlXc0zs81dgwnQQc4 Website: integralsforyou.com/integration-methods/integration-by-substitution 2. Trig substitution: Playlist on KZbin: kzbin.info/aero/PLpfQkODxXi49OUGvTetsTW61kUs6wHnzT Website: integralsforyou.com/integration-methods/integration-by-trig-substitution 3. Weierstrass substitution (t=tan(x/2)): Playlist on KZbin: kzbin.info/aero/PLpfQkODxXi4-8kbKt63rs1xwo6e3yAage Website: integralsforyou.com/integration-methods/integration-by-weierstrass-substitution Thanks for asking! I hope my answer helped you! 💪

[@gabyshavit8539]

@@IntegralsForYou now I know for the next time thank you so much!

[@IntegralsForYou]

@@gabyshavit8539 My pleasure! Enjoy the channel! ❤

[@kouroshA.Z988]

1/2 ln|x^2+a^2|+c

[@IntegralsForYou]

Are you sure? 🙃

[@ShawnGamerTV]

@@IntegralsForYou is it because if it was x on top instead of 2x it would be ln|x^2+a^2/x^2-a^2|?

[@Gust52]

Can you make a video covering all cases of the integral 1/(ax^3+bx^2+cx+d) ,i.e., 1/cubic ? Just like u did for the quadratic one 4 yrs ago.

[@IntegralsForYou]

Hi! The integral of 1/(ax^2+bx+c) had, fortunately, only 3 cases depending on the value of b^2-4ac. In the case of 1/(ax^3+bx^2+cx+d) we need to know how many solutions can be depending on the values of a,b,c,d and I'm afraid there are a lot of cases... Do you know where I can find the list of cases? Thank you! ❤

[@Gust52]

@@IntegralsForYou Sure - since the cases largely depend on the discriminants, you can find info regarding it and many other things related in the following sections of the *Wikipedia page on the Cubic equation* : 1) *Discriminant* 2) *Nature of the roots* 3) *Multiple root* 4) *Characteristic 2 and 3* 5) *Cardano's formula* 6) *General cubic formula* I hope this helps and ty for spending some of your precious time on my request 😊

[@IntegralsForYou]

@@Gust52 Wow, I'll take a look! The video is on my TO DO list now, but I think I will have to work a lot for this integral 😉💪

[@armanraj8286]

Bruh some questions from Indian ncert book❤

[@IntegralsForYou]

Hi! Do you know where I can find the questions? If I have the list of integrals that we find in the ncert book I can do a playlist about it 💪

[@Gust52]

@@IntegralsForYou U can simply search for "NCERT class 12 maths pdf", then u'll get NCERT's website, click on that, then u'll see the PDFs chapter-wise. U can download the one for integrals. For advanced questions, u can search for "NCERT Exemplar PDFs"

[@gelbkehlchen]

Solution: ∫2x/(x²+a²)*dx = ------------- Substitution: u = x²+a² du = 2x*dx ------------- = ∫du/u = ln|u|+C = ln|x²+a²|+C Checking the result by deriving: [ln|x²+a²|+C]’ = 2x/(x²+a²) everything okay!

[@IntegralsForYou]

Thank you! ❤

[@GrannyKiller6245]

Bruh! Take some advanced level integration question based on leibniz theorem.

[@IntegralsForYou]

haha maybe one day...! 💪