Substituting x = tan(u) and then taking u --> pi/4 - u is equivalent to the substitution t = (1 - x)/(1 + x) (which implies x = (1 - t)/(1 + t)) since tan(pi/4 - u) = (1 - tan(u))/(1 + tan(u)) = (1 - x)/(1 + x) = t. Knowing this, using the substitution x = (1 - t)/(1 + t) and exploiting the beautiful symmetry within the integral is undoubtedly the most elegant solution in my mind, since it requires one less substitution and we don't have to venture into the trig world. Feynman's trick is always cool tho. I've never heard anyone call the substitution x = (1 - t)/(1 + t) the Weierstrass substitution, I thought that was just t = tan(x/2). That substitution is actually awesome tho, I used it to solve another integral on your channel by symmetry, the integral from 0 to infinity of (x - 1)/(sqrt(2^x - 1)*ln(2^x - 1)) w.r.t. x. I think a general case of this type of rational function substitution could be very useful for evaluating various definite integrals but I have yet to look into that.
It's been a long time since I was in school and had to do this kind of math, but it is so cool seeing someone who has a natural instinct for it showing all the cool tricks. Makes me feel young again. Thanks for your videos.
Great video. Could you do a video about the Weierstrass substitution itself? E.g., explaining the link to tan(x/2), how it was discovered, how to identify when to use it, etc.
subtituting x = tan(u) and combining 1 + tan(u) to form (sin(u) + cos(u))/cos(u) allows you to apply the harmonic addition theorem for the numerator and to split the integral up by log properties. afterwards, you can apply log properties once again to pull out pi*ln(2)/8 and you're left with two integrals that differ by a parameter. to deal with this you can apply "king's property" and even function properties to arrive at pi*ln(2)/8.
No contour Integral is like the Re(the complex man)... Man weistrauss subs are awesome! I should look over it's "rule of thumb" in regards in how to use it.
Hi Kamal. I’ve been following your integrals enthusiastically but I can’t access your descriptions! I can’t even see how to describe or see a list of the integrals you have solved. I’d like to see a proof of Euler’s reflection formula which doesn’t involve contour integration. This would imply integrating x^(n-1)/(x+1) without contour integration. Is this possible and, if so, how?
![Integrating using t=tan(x/2) substitution - [The Weierstrass substitution]](http://i.ytimg.com/vi/QCAtgZRelno/mqdefault.jpg)
![I.N "HALLUCINATION" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/n5B5q1Hwt_U/mqdefault.jpg)
Substituting x = tan(u) and then taking u --> pi/4 - u is equivalent to the substitution t = (1 - x)/(1 + x) (which implies x = (1 - t)/(1 + t)) since tan(pi/4 - u) = (1 - tan(u))/(1 + tan(u)) = (1 - x)/(1 + x) = t. Knowing this, using the substitution x = (1 - t)/(1 + t) and exploiting the beautiful symmetry within the integral is undoubtedly the most elegant solution in my mind, since it requires one less substitution and we don't have to venture into the trig world. Feynman's trick is always cool tho. I've never heard anyone call the substitution x = (1 - t)/(1 + t) the Weierstrass substitution, I thought that was just t = tan(x/2). That substitution is actually awesome tho, I used it to solve another integral on your channel by symmetry, the integral from 0 to infinity of (x - 1)/(sqrt(2^x - 1)*ln(2^x - 1)) w.r.t. x. I think a general case of this type of rational function substitution could be very useful for evaluating various definite integrals but I have yet to look into that.
It's been a long time since I was in school and had to do this kind of math, but it is so cool seeing someone who has a natural instinct for it showing all the cool tricks. Makes me feel young again. Thanks for your videos.
Was looking for this, thank you.
Great video. Could you do a video about the Weierstrass substitution itself? E.g., explaining the link to tan(x/2), how it was discovered, how to identify when to use it, etc.
subtituting x = tan(u) and combining 1 + tan(u) to form (sin(u) + cos(u))/cos(u) allows you to apply the harmonic addition theorem for the numerator and to split the integral up by log properties.
afterwards, you can apply log properties once again to pull out pi*ln(2)/8 and you're left with two integrals that differ by a parameter. to deal with this you can apply "king's property" and even function properties to arrive at pi*ln(2)/8.
but the feynman integration one is pretty cool!
Fantastic
BRO IS STILL CALM IN HERE
No contour Integral is like the Re(the complex man)...
Man weistrauss subs are awesome! I should look over it's "rule of thumb" in regards in how to use it.
Great video Kamal!
How about writing 1/(1+x^2) as a geometric sum and the do integration by part (using polynomial division)? 😊
In my opinion, you take the win for coming up with the ways to solve it.
Hi Kamal. I’ve been following your integrals enthusiastically but I can’t access your descriptions! I can’t even see how to describe or see a list of the integrals you have solved. I’d like to see a proof of Euler’s reflection formula which doesn’t involve contour integration. This would imply integrating x^(n-1)/(x+1) without contour integration. Is this possible and, if so, how?
Here's a proof for the reflection formula (2nd half of the video):
ruclips.net/video/5VE5kJUJFE0/видео.html
In what device your making these videos ?
Tablet ? Or phone ? Or computer ?
Would like to see a solution using contour integration
Coming up in a few days
I like the second approach because it is the easiest way to calculate it in my opinion
Can it be solved by contour integration? (for Feyman's trick,let I(a)=int[ln(1+ax)/(1+x^2)].It is also possible.
No links in the description!
Fixed it
the answer should be п*ln(4)/8
Standard integral in iit