HAVE YOU DONE THIS WITHOUT INTEGRATIONS BY PARTS?

[@moth5799]

You can evaluate the indefinite integral with a matrix, since the span of xsinx, xcosx, sinx and cosx is closed under differentiation and integration. Super quick and can be applied to any linear combination of the functions, it's great.

[@gayansamarasekara]

Absolutely. It’ll be shown on a future video. Thanks for the idea. 👍🏻

[@mimorouidjali5487]

Can you do a video about solving integrals with matrices, Polar coordinates, complex num?

[@carultch]

@@mimorouidjali5487 Here's an example of how you'd solve an integral with a matrix. Given: integral (5*x^2 + 10*x + 7)/sqrt(x + 1) dx Assume the solution has the form, called your Ansatz: (A*x^2 + B*x + C)*sqrt(x + 1) The reason we can assume this, is that each power increases by 1, and the square root term is maintained. As long as the solution isn't a logarithm or inverse trig, there will be an algebraic solution that is a product of a polynomial and a form of the original square root. If it involves a log or inverse trig, you'll get a degenerate matrix. Take the derivative of the Ansatz: 1/(2*sqrt(x + 1)) * (A*x^2 + B*x + C) + (2*A*x + B)*sqrt(x + 1) De-rationalize the denominator of the second term, to get it to look like the original integral: 1/(2*sqrt(x + 1)) * (2*A*x + B) + (A*x^2 + B*x + C)*sqrt(x + 1) Expand and gather like terms: (5/2*A*x^2 + (2*A + 3*B)*x + 2*B + C))/sqrt(x + 1) Now we can match coefficients to the original integral: 5/2*A = 5 2*A + 3/2*B = 10 B + C/2 = 7 Which we can represent as a matrix equation: [5/2, _ 0, _ 0] _ [A] _ [5] [2, _ 3/2, _ 0] * [B] = [10] [0, _ 2, _ 1/2] _ [C] _ [7] And with our favorite matrix solving method, we can find the solution: A = 2, B = 4, C = 6 Thus the solution is: (2*x^2 + 4*x + 6)*sqrt(x + 1) + K

[@josjos1847]

Great work, I love who you solve integrals without using the usual methods and we learn from it

[@gayansamarasekara]

Thank you so much for your wonderful comments....! I will try to make more videos.

[@adiramrakhani]

Great video, just subscribed after watching all your older videos too there are some excellent tricks here

[@gayansamarasekara]

I am glad you watched my videos. Thank you so much for your encouraging comment and subscription.

[@johnrm9]

Oh, The Famous Kings Property!

[@gayansamarasekara]

The famous Kings….! ✅

[@samueldeandrade8535]

You could do the general case ...

[@gayansamarasekara]

Yes, a generalization of the theorem can be found here: CAN YOU EVALUATE THIS DEFINITE INTEGRAL? kzbin.info/www/bejne/gaHWpaFvepaGm9U

[@simongross3122]

That is a really clever trick, thank you. Can you please point me to the video where you show how the identity you used is derived?

[@gayansamarasekara]

Thank you so much. I think I proved the initial version here: kzbin.info/www/bejne/horZeadteatlmdU

[@simongross3122]

@@gayansamarasekara Thank you. That also is clever. It occurs to me that for any definite integral, so long as neither limit of integration is infinity, then we can always make a substitution such that we have an integral from zero to something and then we can further apply this wonderful trick. It's mathemagical :)

[@gayansamarasekara]

@@simongross3122 Exactly....! Also, when we are a little too bored, we can fix the genes of the theorem, and have it ready for any asymmetric limits, such as the one discussed here: kzbin.info/www/bejne/gaHWpaFvepaGm9U, to be more mathemagical :) Thank you for your nice mathemagical comment....!

[@simongross3122]

@@gayansamarasekara Haha my pleasure. I came across that particular phrase a long time ago when I worked in the IT industry. I'm not actually a mathematician although I have always had a keen interest, more in the philosophy than the practice. I'd rather be a mathemagician :)

[@simongross3122]

@@gayansamarasekara I watched that video and I was not disappointed. I was expecting you to have a theorem that went Integral from limits a to b being replaced by integral limits from 0 to something (perhaps b-a), but you surprised me. :) I am pretty sure this also can be done.

[@gentlemandude1]

Where can we find the video with the complete proof? Thanks.

[@gayansamarasekara]

Please check this: kzbin.info/www/bejne/gaHWpaFvepaGm9Usi=iRS3L4m4DTcgllvh

[@psycosingershayar]

Do this instead Special case for such questions where we hsve xf(x) if f(x). Does not change after applying kings rule the. We can remove x by x=(upperlimit + lowerlimit)/2 and then Integra f(x)

[@gayansamarasekara]

Good Comment on Kings Rule. This video discusses an example from a popular form of problems found in calculus 1 classes, as a special case of the property, where the lower limit is zero.

[@rajrajnish3136]

Please make a video about yourself.

[@gayansamarasekara]

Thank you, I will keep that in mind. Basically, I am a professor at k-state, USA, I teach math and statistics classes and do research. I'm thinking to make videos for this channel to help the students in: college algebra, trig and calculus 1, 2, 3 courses taught for non-math undergrad majors of STE(M) fields of the US universities.