My Most Controversial Integral

a while ago I made a video on this integral if you can call it that let's be clear from the beginning this is not exactly a welldefined expression integrals are typically defined with the differential not in the exponent and so in my original video I played around with these differentials in an admittedly dubious way to get an answer and while on this channel I do like to sort of push the balance of what's mathematically acceptable there is a little bit better way to address this expression if we want to integrate this we can't have the differential in the exponent and a great way to deal with this traditionally is by using logarithms if we insert an e to the Ln here we can use that famous property of logarithms to drop the exponent down front as a coefficient now already this is getting a little bit weird because that exponent is a differential it's an infimal nearly zero so does this even make sense well let's just keep going with it now how do we deal with this let's use the series definition of e to the X it's just not X in the exponent here it's this natural log DX and so things are getting a little bit complicated however if we start expanding this some writing it out term by term I mean there are an infinite number of them but it is something we can sort of deal with we'll use the usual definition that zero power means this quantity is one and we'll keep adding on natural log DX to increasingly large powers and here's the purpose of that minus1 originally we'll just cancel out this 1 minus one from the front and here's where we need to think a little bit more these DXs these infinim Mals are almost like zero they're sort of like infinitely close to zero so squaring them or better yet a higher power of them is almost definitely zero like even closer closer to zero so close that we will treat all of these DXs to a power higher than one basically as zero meaning all of these other terms are just zeroing out and so our integral reduces to just the anti-derivative of natural log X DX which is something that we can definitely solve you can just use a little bit of integration by parts here let U be natural log X DV will be DX and we'll just follow the integration by parts formula and what's very interesting is this is the exact same result that we got in my previous video using the more dubious mathematical methods if you thought this weird expression was interesting I want to share something equally intriguing with you the sponsor of today's video brilliant.org if you're like me and love getting into the nuts and bolts of things you'll definitely appreciate what brilliant has to offer brilliant isn't just like another online learning platform it's a Hands-On Adventure into the world of math science programming and AI what sets brilliant apart is their approach to learning by doing instead of passively watching videos you're actively solving real problems which makes the learning process so much more effective and engaging now speaking of AI the one course I'm really interested in is Brilliance how llms work this course takes you on a deep dive into the mechanics of language models like the ones that are so popular today you'll get to experiment firsthand with how these models build vocabulary make predictions and how they can even be fine-tuned to produce different types of output it's a great way to see theory in action and really grasp how training models like these work today the best part you can start learning with brilliant today free for a full 30 Days by visiting brilliant.org brithemathguy plus you'll also get 20% off an annual premium subscription and if you want to see the other slightly more dubious way I've tackled this Express question check out the video here I'll see you in that one

Channel: BriTheMathGuy

Views: 23,281

Keywords: math, maths

Id: AiLVpubwZ48

Length: 4min 35sec (275 seconds)

Published: Mon May 06 2024