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! ❤
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.
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! ❤
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Already did
@@BK2410 Love you! 🥰
@@IntegralsForYou love you too!
This channel will save me through this semester, thanks a lot
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Nice job!
Thanks! ❤❤
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!
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! ❤
@@IntegralsForYou Much appreciated! Your channel is very helpful and i thank you for hard work!
@@ahmedjouda8793 Thank you! ❤ It is my pleasure! ☺
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.
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! ❤
@@IntegralsForYou Thank you, I did end up figuring it out.
@@DoubleDabbles Good job, then! 💪
Integral of 1/[sqrt(x)(1+2lnx)]
I hope you do it soon !!
Hi! I think it is a non-elementary integral, it cannot be expressed in terms of finite standard functions...
36// 18.06.24. Да , я тоже решал его по частям ( by parts).
Good job! ❤
Hi. Please do this integral :
Integral [t/(t^2 +x^2) ] dt
PLEASE FAST . Thankyou❤️❤️❤️
Hi! I did the video here: Integral of x/(x^2+a^2) dx = th-cam.com/video/4qC9_VMRLZw/w-d-xo.html ❤️
Super
Thank you! ❤