Needs some more tidying up at the end, Michael. Any ln(1/something) needs replacing with -ln(something) and any log roots need replacing with half logs. You’re getting marks docked for that. ln(1-half.ln(2)) - ln(half.ln(3) - ln(2)+third.root3)
add in the numerator cot(x) and then subtract it, split the integral into 1-(cot(x)-csc^2(x))/ln(sinx) and notice that the numerator is the derivative of the denominator, so the substitution u=ln(sinx) gets it done. Edit: this is basically what you did, but without substituting ln(sinx)=f(x)
Wolfram alpha gives us the indefinite integral of that : ∫(csc²(x) + ln(sin(x)))/(cot(x)+ln(sin(x)))dx = x + log(sin(x)) - log(cos(x) + sin(x) log(sin(x))) + C
This video has made me wonder (phps heretically) - is there a point to the functions csc, sec, cot - and if you really want to go there... tan? Do they really justify a label?
@@jacemandt Fair, but I mean why introduce new labels when: tan = sin/cos, sec = 1/cos, they are not fundamental in any way - they feel like a hangover....
It's not that f'(x)/f(x) = ln(f(x)). Rather the RHS is an antiderivative of the LHS. You can see this by taking the derivative of ln(f(x)) using the chain rule.
FUCK qhybajdnh8wnin gods name wiuld.anyone noticeable numeratornisnjust f if x minus f orime..I don't see angine seeing that at all just a lucky dirty trick and if you missnyoundont see it..surelynthere must be a logical smarter way..
Michael has a google form for people to suggest problems (in the description box of the video). Someone spams the form with a bunch of problems that end up being interesting enough for Michael to make videos with them. Their pseudonym on the form is “integral suggester” or I guess “problem suggester” now. I don’t think he knows who the person actually is.
The animation at 1:35 looks so nice! You should do it more often :)
I love the transparency effect as well 🥰
Needs some more tidying up at the end, Michael. Any ln(1/something) needs replacing with -ln(something) and any log roots need replacing with half logs. You’re getting marks docked for that.
ln(1-half.ln(2)) - ln(half.ln(3) - ln(2)+third.root3)
5:58
That was a nice simple problem(in the grand scheme of things). Thank you
loving these trigonometric integrals lately, keep it up!
Substitution at its finest! NIce job.
A case of follow your instincts.
Thank you, professor.
Cool problem. Thanks for the inspiration!!
add in the numerator cot(x) and then subtract it, split the integral into 1-(cot(x)-csc^2(x))/ln(sinx) and notice that the numerator is the derivative of the denominator, so the substitution u=ln(sinx) gets it done.
Edit: this is basically what you did, but without substituting ln(sinx)=f(x)
Wolfram alpha gives us the indefinite integral of that :
∫(csc²(x) + ln(sin(x)))/(cot(x)+ln(sin(x)))dx
=
x + log(sin(x)) - log(cos(x) + sin(x) log(sin(x))) + C
nice video!:)
This video has made me wonder (phps heretically) - is there a point to the functions csc, sec, cot - and if you really want to go there... tan? Do they really justify a label?
Tan (at least) deserves a label because if a line goes through the origin, its slope is calculated with the tangent function.
@@jacemandt Fair, but I mean why introduce new labels when: tan = sin/cos, sec = 1/cos, they are not fundamental in any way - they feel like a hangover....
Suggest me a book or pyq of the examination where I get such questions equivalent or greater level to jee advanced
I never knew that f'(x)/f(x) = ln(f(x)). Never was taught such a thing. Thanks, but where does it come from?
It's not that f'(x)/f(x) = ln(f(x)). Rather the RHS is an antiderivative of the LHS. You can see this by taking the derivative of ln(f(x)) using the chain rule.
@@allanjmcpherson oh yeah you're right, thanks. sometimes i'm a bit slow
@@bronchiel no worries. I'm sure we've all been there. I know I have.
FUCK qhybajdnh8wnin gods name wiuld.anyone noticeable numeratornisnjust f if x minus f orime..I don't see angine seeing that at all just a lucky dirty trick and if you missnyoundont see it..surelynthere must be a logical smarter way..
Approximately 0.672, for the record.
Hi Dr.!
I’m new here. Who’s the “favorite problem suggested”?
Michael has a google form for people to suggest problems (in the description box of the video). Someone spams the form with a bunch of problems that end up being interesting enough for Michael to make videos with them. Their pseudonym on the form is “integral suggester” or I guess “problem suggester” now. I don’t think he knows who the person actually is.
@@stephenbeck7222 TY!!! you rock dude
Crushed it! What an ugly number though
yo dawg I heard you like natural logarithms
@@kostasbr51 would love to see a proof
الفكرة رائعة و الخبير هو من سيلاحظها.
تحية لكم
asnwer=1(dx+xd)(sin-sin) 🤣🤣🤣🤣🤣