As soon as you got to the step in which you only had a trig in the exponent, and a composite of two trig functions, I realized (with the help of the title) that this is like one of my favorite qualifying exam problems, just slightly more complicated. With that in mind, you could have used Cauchy's Integral Theorem after one more change of variables after that same step, say letting z=e^{ix} and the problem becomes a contour integral. Of course, the two methods are equivalent by the same theorem - many such cases in complex analysis!

Or you could use Euler's formula at the getgo and write the cosine as the real part of the exponential: cos (1/(1+x^2) = Re exp [ i /(1+x^2) ], and bring it into the exponential of x/(1+x^2): I = Re integral from (- inf) to (+ inf) exp [ (x+i)/ (1+x^2) dx/(1+x^2) Since 1+ x^2 = (x+i)(x-i), the exponential becomes exp [ 1/(x-i) ]. The other factor becomes 1/(x+i)(x-i), with poles at z= +- i. Closing the contour below the x-axis to enclose the pole at z=-I, and to avoid the exponential pole at z= +I, you get from Cauchy's Theorem I = Re (-2 pi i) /(-2i) exp [ 1/(-2i) ] = Re pi exp [ I/2 ] = pi cos (1/2)

If we consider regions like semicricle with centre at origin and use cauchy residue formula. This entire thing will get simplified. Many of the integrals can be evaluated using cauchy residue formula. Feynmans techniques, fubini theorem can sometimes be tedious. When it comes to complex residue theorem we are hundred percent sure we will get the final answer. We have to choose the region properly in a clever way

Hi, "ok, cool" : 3:17 , 6:33 , 9:27 , 10:48 , "terribly sorry about that" : 3:42 , 5:25 , 9:13 .

Wow Really satisfying indeed!!

our boy Kamal bringing out another BANGER of a video!!!

Who else used contour integration ?

Thank you for this pretty solution.

Hey Friend 😊. I found something terrible. Go to 2:24 Int - pi/2 to pi/2 e^(sinxcosx) cos(cos^2 x) dx Int - pi/2 to pi/2, e^(1/2 sin(2x)) cos(1/2 + 1/2 cos2x) dx Let, I(a) = int - pi/2 to pi/2, e^(a/2 sin(2x)) cos(1/2 + a/2 cos2x) dx I'(a) = 1/2 int - pi/2 to pi/2, e^( a/2 sin(2x)) sin( 2x - (1+acos2x)/2 ) Convert sin(***) into polar form, I'(a) = 1/2 int -pi/2 to pi/2 e^(a/2 sin2x) img{ e^(2ix) * e^-i( (1+acos2x)/2 )} dx =1/2 img int - pi/2 to pi/2, e^(a/2 sin2x) e^(2ix) e^-i(( 1 +acos2x) /2) =1/2a img int - pi/2 to pi/2, e^-i/2 e^{-ia/2 ( e^2ix)} (-ia/2) (2i) e^2ix dx Take - ia/2 e^(2ix)= z we shall find it's just intgrl 1/2a img (e^z dz) = 1/2a img [ e^{ - ia/2 (e^2ix)}] put limits (pi/2, -pi/2) = 1/2a img [ e^{ - ia/2 e^ipi} - e^{ - ia/2 e^-ipi}] =0 Cuz, We know e^ipi=-1, and e^-ipi=-1 too So, I'(a) =1/2 img(0) =0 I(a) =C =I(0)=pi cos(1/2)=I(1) =I(💙)=I(kamaal) For every real a, I(a) has a fixed value of pi cos(1/2)

We can use complex analysis contour integrals.

الله يرحم والديك

Can you tell me a book which contains these types of integrals along with solutions?

Oh look: It's the "Tangent double half-angle substitution". :D

Can we use the SE-METHOD ?

I'm guessing the substitution u=1/(1+x^2) is needed here? 😂

While I am not so froggy as to seek out integrals to handle on my own, I do appreciate the ability to see a lot of the techniques used just by eyeballing the function in the thumbnail. Some of your substitutions are still wild, but I've experienced far fewer "how the hell did he come up with that" moments. I guess having watched your channel from the start has done something. (well, minus the physics stuff... with apologies)

Amazing ❤

I like to try these having looked only at the thumbnail. I also only like to try theorem if I can use contour integration. This is what I did and I did in fact get your solution. The integrand simplifies to Re exp(1/(z+i))/(z**2+1). Close the contour in the lower half plane and use the pole at -i. I think this is all valid as the integrand vanishes like 1/z**2 on the big semicircle. What do you think?

Make the denominator another integral in terms of e^x take real part to make cosx in terms of e^x and proceed

Wtf there’s no title picture Edit : I reloaded and there now is. My pc bugging

Stopped the video at 0:16 entered into cheat mode: Off to Desmos to graph the thing to see what it looks like. Performing the absolutely obvious u-substitution. Restarted the video, and it is the wrong u-sub. I had u = x^2 + 1

AMAZİNG

@10:45 why does the exponential term simplify to 1?

Wow 😮

Ho fatto come te,ma poi mi sono bloccato su I1,perché avevo indici di integrazioni diversi..ho sistemato gli indici di integrazione e risulta πcos(1/2)

I am lost at 5:40 for a moment. Where did isin go? Why only real? I need complex analysis(

I thought I was effective at integrals, but you've shown be greatly otherwise.

I am a highschooler and this scares me 😣

Bro I have another challenge for you! Σ (0 to ∞) ( (r²)/(2^r) ) Give me the correct answer

asnwer=1x