Integral of x*arctan(2x)

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• 🧑‍🔧 Integration by parts
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🚶 𝐒𝐭𝐞𝐩𝐬
00:00 Parts: Integral of u dv = uv - Integral of v du
00:11 Choose u and dv
00:23 Differentiate u to get du
00:44 Integrate dv to get v
00:50 Substitute u, v and du
01:16 Simplify expression
01:44 Multiply and divide by 4 and add +1-1 on the numerator
02:21 Split into two integrals
03:00 Simplify and multiply and divide by 2
03:45 Integrate 1 and 2/(1+(2x)^2)
04:08 Rewrite expression
04:29 Add integration constant +C
04:41 Final answer!
04:46 See more!
#integralsforyou #integrals #integration

[@IntegralsForYou]

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

[@BK2410]

Already did

[@IntegralsForYou]

@@BK2410 Love you! 🥰

[@BK2410]

@@IntegralsForYou love you too!

[@Tod_x]

This channel will save me through this semester, thanks a lot

[@IntegralsForYou]

Glad to hear that! ❤ Please, share the channel with your friends and colleagues, it will help my channel grow! Thank you! ❤❤

[@MikeT10101]

Nice job!

[@IntegralsForYou]

Thanks! ❤❤

[@mamadetaslimtorabally7363]

Super

[@IntegralsForYou]

Thank you! ❤

[@DrewMcIntyre.]

Hi. Please do this integral : Integral [t/(t^2 +x^2) ] dt PLEASE FAST . Thankyou❤️❤️❤️

[@IntegralsForYou]

Hi! I did the video here: Integral of x/(x^2+a^2) dx = kzbin.info/www/bejne/aqKmapKMgrd_kNk ❤️

[@SamSokha329]

Integral of 1/[sqrt(x)(1+2lnx)] I hope you do it soon !!

[@IntegralsForYou]

Hi! I think it is a non-elementary integral, it cannot be expressed in terms of finite standard functions...

[@Snow_Leopard_Uncia_uncia]

36// 18.06.24. Да , я тоже решал его по частям ( by parts).

[@IntegralsForYou]

Good job! ❤

[@DoubleDabbles]

Thank you, watching your videos has helped me a lot. Is there any chance you could do the integral of x^5/(1-x^2)^3/2. I seem to get very close using techniques I already know but everywhere I look is telling me I'm doing something wrong.

[@IntegralsForYou]

Hi! Of course! Here you have the solution: Integral of x^5/(1-x^2)^(3/2) dx = = Integral of (x^2)^2/(1-x^2)^(3/2) x*dx = Substitution: u = 1-x^2 ==> x^2 = 1-u du = -2x*dx ==> du/-2 = x*dx = Integral of (1-u)^2/u^(3/2) du/-2 = = (-1/2)*Integral of (1-u)^2/u^(3/2) du = = (-1/2)*Integral of (1-2u+u^2)/u^(3/2) du = = (-1/2)*Integral of ( 1/u^(3/2) - 2u/u^(3/2) + u^2/u^(3/2) ) du = = (-1/2)*Integral of ( u^(-3/2) - 2u^(-1/2) + u^(1/2) ) du = = (-1/2)*( u^(-1/2)/(-1/2) - 2u^(1/2)/(1/2) + u^(3/2)/(3/2) ) = = (-1/2)*( -2u^(-1/2) - 4u^(1/2) + (2/3)u^(3/2) ) = = u^(-1/2) + 2u^(1/2) - (1/3)u^(3/2) = = (1/3)u^(-1/2)(3 + 6u - u^2) = = (1/3)(1-x^2)^(-1/2)(3 + 6(1-x^2) - (1-x^2)^2) = = (1/3)(1-x^2)^(-1/2)(3 + 6 - 6x^2 - 1 + 2x^2 - x^4) = = (1/3)(1-x^2)^(-1/2)(8 - 4x^2 - x^4) = = (1/3)(1/sqrt(1-x^2))(8 - 4x^2 - x^4) = = (-x^4 - 4x^2 + 8)/3sqrt(1-x^2) + C I don't have time to do the video right now but one day I will, it is on my TO DO list now! Hope my comment helped you! ❤

[@DoubleDabbles]

@@IntegralsForYou Thank you, I did end up figuring it out.

[@IntegralsForYou]

@@DoubleDabbles Good job, then! 💪

[@ahmedjouda8793]

1:44 As a beginner, how am i supposed to figure out when to add the +1 and -1 expression? Also i really appreciate your work!

[@IntegralsForYou]

Hi! I wrote a little bit about this topic in my website: integralsforyou.com/frequently-asked-questions#why-do-we-add-and-subtract-constant-numerator The short answer to your question would be: when the degree of the polynomials on the numerator and denominator are equal 😉 Thanks for your comment! I would be very happy if you could help me grow my new channel about derivatives by subscribing 👉 www.youtube.com/@DerivativesForYou?sub_confirmation=1 Thank you! ❤

[@ahmedjouda8793]

@@IntegralsForYou Much appreciated! Your channel is very helpful and i thank you for hard work!

[@IntegralsForYou]

@@ahmedjouda8793 Thank you! ❤ It is my pleasure! ☺