Integral of e^x/x vs. integral of 1/ln(x)

We will integrate e^x/x and integrate 1/ln(x). These are non-elementary integrals but we can use special functions to integrate.
Exponential integral function Ei(x), en.wikipedia.org/wiki/Exponen...
Logarithmic integral function li(x), en.wikipedia.org/wiki/Logarit...
Subscribe for more math for fun videos 👉 @blackpenredpen
💪 Support this channel, / blackpenredpen
🛍 Shop math t-shirt & hoodies: bit.ly/bprpmerch. (10% off with the code "WELCOME10")
🛍 I use these markers: amzn.to/3skwj1E

[@MuPrimeMath]

0:07 agreed!

[@allaincumming6313]

The logarithmic integral is very very interesting, since it can be considered as an incredibly approximation to the Prime number counter function, π(x), where xε|R and π are the number of primes equal or less than x.

[@almightyhydra]

I encountered Li(x) in the context of number theory at university, where Li(x) gives (roughly) the number of primes less than or equal to x.

[@mathqed5114]

why is math so beautiful

[@erichschmidt1328]

"Let me write it down here for you, guys..." Thank you for these words. It's a real pleasure to watch your videos.

[@patriciomejia1114]

BPRP: "Let's do some math for fun." Other people: "I'm traumatized." Me: "I'm excited."

[@blackpenredpen]

Yo, Andrian!

[@natealbatros3848]

Can we use series expansion to get the non elementary functions?

[@alexdemoura9972]

"... more functions we have, more Integrals we can do, more pleasure that we can have... in our lives..., isn't it?... YES!!!" LOL, in my opinion the best intro of this channel ever, even more than "YAY!!!"

[@JITENDRAKUMAR-hj9ht]

√sinx dx integration without limit

[@shandyverdyo7688]

Hiiii bprp.... The coolest integraler .. Could you make a video about integral imvolving hypergeometric function??? I just curious. Thanks.

[@angelmendez-rivera351]

matan guedj You would not be able to simplify the series.

[@Dlmlai7255]

Best approach!! I'm studying from iit Bombay, Mumbai, India

[@ritwikbilgi4168]

Bombay is another name for Mumbai............ 😑

[@prakashdubey9884]

Kya padh rahe ho??

[@MartinPuskin]

Another cool thing: Li(x) (the big brother of li(x) is a very good approximation of π(x), the prime counting function i.e. if you want to know how many primes are smaller than 10000000 you can approximate with Li(10000000).

[@roderickwhitehead]

Thank you for covering these.

[@jlsmatejuanluisramirez]

You are so good! I've learnt a lot with your videos

[@josephymenchu]

Alright that was impressive. Love your videos.

[@plaustrarius]

I love it awesome work thank you so much!!

[@VibingMath]

"Dont let w bother u" 😂 Please keep uploading more of this series of videos! Thank you bprp! 😊

[@AjourChannel]

Such an inspiring word

[@efeguleroglu]

7:52 I laughed out loud.

[@pranjaldas1762]

Thank you bprp It was helpful

[@ffggddss]

Very nice! (But I thought you were at least going to mention how Li(N) is a better estimate of the number of primes < N than the customary rule-of-thumb, N/lnN . . . however, I see that you've given the link for the Wikipedia article on li(x), so it's covered there.) As always...That's Fun! Fred

[@theonestemuragijimana3835]

Thank you very much. Your contribution is awesome !

[@alekseikhalin5808]

How about "Don't let W double-bother you"?

[@tsheringdorjigamming36]

I really love your lecture and teaching lesson. Thank you for your mathematics lesson.. 😍

[@dwaynenobleza2912]

You deserve more subscribers!

[@willyou2199]

Li(x) = Ei(ln(x))

[@edgarb.6187]

I was going to ask what the integral of log base x of n was. But then I realised it be the same as the integral of ln(n)/ln(x) where ln(n) can be factored out so you answered this question!

[@afafafaf9360]

Thank youuuu

[@chuldichilli7832]

can u integrate this? ln^2(sinx) dx from 0 to pi/2 i think it's far too difficult

[@justabunga1]

The answer is non-elementary. You have to do it on your graphing calculator and enter it as fnInt((ln(sin(x))^2,x,0,pi/2) and you get the approximation of 2.047. The exact answer if you see from Wolfram Alpha is (pi^3+3pi(ln(4))^2)/24.

[@davidedmundtochi5228]

Prof you are funny😂 , the more pleasure we can have in our lives

[@lasithathilakarathne3205]

your smile makes it a whole lot better🥰

[@michaelpatterson1300]

I think the reason they put it in that worse form is that if you use the 2nd Ei(x) form it has a singularity and you'd need to put pv in front of the integral to take the cauchy principle value

[@anymeskywalker5463]

Thankyou for making this vedio I had commented yesterday about explaining Ei(x)

[@sanjaybalaji9206]

Even I had commented that...thank you sir

[@blackpenredpen]

Thanks for the comments. This video was actually in the description in the previous video. And I just made it public today. : )

[@anymeskywalker5463]

💝😊

[@Patapom3]

Amazing!

[@angelmendez-rivera351]

You should also have explained that the integral definition of Ei(x) is only well-defined for x < 0, and that for x > 0, one needs the Cauchy principal value as well. An alternative definition of the exponential integral is the function E(1, z), or E1(z), which is given by the integral from z to ♾ of e^(-t)/t, which approaches Γ(0+) as z -> 0, which is ♾. This definition works for all complex numbers, except for the non-positive real numbers. Its related to Ei(x), because for real positive x, - E1(x) = Ei(-x), using the Cauchy principal value of the integral definition of Ei(x).

[@chiragijain2936]

@blackpenredpen can u solve this? Indefinite Integration of --> [{sin^2 (x)}*{cos^2 (x)}]/{sin^3 (x) + cos^3 (x)}^2 [sin square x * cos square x] divided by[sin cube x + cos cube x] square Whole square in denominator only

[@YASH-jz4sh]

Very easy hai

[@justabunga1]

The answer is -1/(3(tan(x))^3+3]+C.

[@ozzymandius666]

Math is so elegant.

[@user-nw7oj3rx9i]

Man made an egg out of the integral of e^x/x

[@goda743]

Next time on bprp: Don't let μ bother you.

[@AliAhmed-mp6df]

What a Coincident! Because I was willing to know about Exponential Integral today. 😏 And I got this Video 😍😍😍😍 @blackpenredpen

[@joanmartinsuarezloaiza2767]

Ali Ahmed me too

[@omograbi]

Don't let "Double u" "Doubt u"

[@eduardobauerbauer947]

Nice video! Is there any applications for these functions?

[@jadegrace1312]

The logarithmic integral roughly approximates the prime counting function

[@angelmendez-rivera351]

Eduardo Bauer Bauer These are widely used in physics.

[@dougm275]

Oops, this isn't the Calc 2 class. Sorry for the interruption!

[@gidobaggerman1139]

i have found the integral of ln(ln(x)) it's: xln(ln(x))-Ei(ln(x))+c . I used u=ln(x) and then integration by parts

[@LauraRodriguez-yn5xz]

I have found that equivalence by substituting e^x=t and then using derivation...I found it much easier...

[@kostantinos2297]

I was wondering, is there a formula to rotate a graph of a function around a given point of coordinates (x,y) at an angle θ? I tried switching to parametric form (t,f(t)) to find the resulting graph in a polar-like manner, however the process was painstakingly long and difficult, making it impossible to finish. Are there any quicker or easier ways to do that?

[@noventium]

If that's your question, then you'll surely like this video: kzbin.info/www/bejne/eIHKomV3qsmie9E

[@kostantinos2297]

@@noventium I've seen this, and it is in fact where I got my question from. I wish to rotate a curve around any given point, not just (0,0). I tried it in a similar manner, but using vectors instead. Pretty neat video, though.

[@SR-kd4wi]

Please help with this problem! A(n+2)=3A(n+1)/2 - A(n)/2 , for n€N. Find limit A(n) for n tends to infinity.

[@chronicsnail6675]

What happens if you use integration by parts for the first integral, I did it and got in the d colum (1/x +1/x^2 + 1/x^3+...) So and if limit n -> infty 1/x^n = 0 then the integrand becomes e^x(1/x+1/x^2+...1/x^n-1)

[@akshatahuja2523]

Do some mit integrals especially 2006 th year

[@winter_c]

How can we elevate these two functions?

[@pianoforte17xx48]

6:22 that's a heavy *W*

[@p.g.wallychopin]

A challenge: Find a real solution for the equation x^x + Ln x = 4, using the W-Lambert function

[@jollardsmathers7804]

can u suggest topics about calculus? to be used in my dissertation

[@jonathangrey6354]

0:10 turn of subtitles

[@machinetool9360]

can we derive general conic section formulas (polar) using calculus ?

[@lukamitrovic7873]

I'm pretty sure you can. Just replace x with rcos theta and y with rsin theta and you should be done :P

[@Harkmagic]

So how do I do anything useful with this? I assume some form of Numerical Analysis, but I feel like that kinda defeats the point of having a function to keep it simple.

[@markzero8291]

I had the same thought.

[@angelmendez-rivera351]

Physics.

[@Harkmagic]

@@angelmendez-rivera351 physics is probably why I would want to do something useful with it, but doesn't really explain how to do it.

[@andywright8803]

What would make this video perfect would be sketches of what these functions look like. (I am watching from a hospital bed and pen and paper stuff is difficult atm)

[@Laff700]

Try doing math in a word document. Do Alt+= to start writing an equation.

[@andywright8803]

@@Laff700ill give that a go. When I leave hospital

[@Laff700]

@@andywright8803 Oh, you're watching it on a phone, not a laptop...

[@andywright8803]

@@Laff700 does alt+ draw a sketch of the graph, or merely allow you to write the equation

[@YourPhysicsSimulator]

Nice

[@DefenderTerrarian]

My Calc2 professor made us approximate 1/lnx definite integral from 1 to 2, by Simpsons. I calculated the fourth derivitave by hand. When we covered it in class, he after looking at it, he said he made a mistake and meant to put lnx from 1 to 2. I was so mad, still am I guess. He's a horrible teacher.

[@MathZoneKH]

i'll try to understand. Mr Prof.

[@bandamkaromi]

WHAT A BEAUTIFUL..

[@benjamingarcia5703]

What about integral of e^(1/x^3)/x^3

[@yashpandey5416]

I got e^x(x-1)^(-1)

[@zenmaureal4139]

Please help... what is integral of (exp-(k-1)x)/x from 0 to infinity

[@somanathdash8143]

Pls make videos on 3d vectors...

[@manla8397]

Ln(0) is also undefined.

[@christosdelivasilis]

Can you show how to integrate e^arctan(x+1)?

[@philipphoehn3883]

has no elementary solution

[@pippoborraccia]

Hi, can you integrate this? (cos(x))^sin(x)?

[@-fai7485]

taylor

[@graywolf2600]

May I ask for polylogarithm functions? :D

[@blackpenredpen]

I plan to cover them.

[@ayushk3870]

Sir plz integrate x^x plz plz

[@cassiel5150]

It is non elementary

[@ayushk3870]

@@cassiel5150 thx

[@hexeddecimals]

What is the limit of Ei(x) as x goes to 0?

[@justabunga1]

Hexed Decimals The answer is -infinity.

[@ferlywahyu342]

Well how with Ei(xi) are you know

[@zeusaurel6714]

don't let dabiu badiu

[@Rajat_Pathania]

It was fun.. but i think m never gonna get my answer now..😂😂😂😂

[@Scott21]

Is this brand new for math? I'm sorry if I sound dumb here, but I have just never seen these before.

[@matrixstuff3512]

It's not new, but it is somewhat obscure. Especially if you don't have a degree in math.

[@Scott21]

@@matrixstuff3512 Thank you again, it really is a first for me and I just wasn't sure if I missed something in one of my calculus courses. Do you have a major in math? If so what is your favorite math class so far? You're awesome by the way.

[@DavidDavid-sd2gd]

Li(x) is actually an approximation of the amount of primes under x

[@kingarth0r]

Kyle W hehehe

[@vishwakalhara6235]

ooh finally found it 😂

[@jeelvanjara1689]

Can we find integral of 1^x???

[@guerfirayene8938]

Can you give me integral of e^(x)tan(x)

[@rachels2cents325]

I did the inverse of the li function.

[@ruthbautista2865]

How do you integrate xtanx?

[@justabunga1]

Ruth Bautista the answer is non-elementary. If you try to do IBP, it would come up as -xln(abs(cos(x)))+integral of ln(abs(sin(x))). According to Wolfram Alpha or others, the answer is (i(2ixln(abs(e^(2ix)+1)))+Li_2(-e^(2ix))+x^2)/2+C if you’re doing the complex solution. Li_2(x) is a polylogarithmic function, which is a non-elementary function.

[@DeepakYadav-ve7pg]

Pls do the integration of root sinx

[@justabunga1]

Deepak Yadav the answer is non-elementary. If you look up Wolfram Alpha or other websites, the answer is 2E((2x-4)/4|2)+C. The E(x) stands for elliptic integral where the notation here as E(x|m) stands for the parameter with m=k^2 where k is an eccentricity if you remember from conic sections. This is NOT a probability notation.

[@pinchus2714]

How do you integrate √x cos(x)?

[@pinchus2714]

@ I mean what's the answer

[@angelmendez-rivera351]

Pinky Cohen The answer is non-elementary, surprisingly.

[@pinchus2714]

@@angelmendez-rivera351 Thanks. I was trying to do it with integration by parts...it didn't work. I got an even more complicated integral at the end.

[@justabunga1]

It’s non-elementary. You can try IBP, it would come up as sqrt(x)sin(x)-integral of sin(x)/(2sqrt(x)). If you look it up, the answer is sqrt(x)sin(x)-(sqrt(pi/2))S(sqrt((2x)/pi))+C. S(x) stands for sine Fresnel integral.

[@srikanthtupurani6316]

1/ln(x) looks so annoying. But if think about it deeply we will be liking it.

[@aneeshsrinivas9088]

what's the easiest nonelementary integral.

[@Kradrice]

Did I see you at Metrotown mall today?!

[@blackpenredpen]

I don’t think so. Where’s that?

[@Kradrice]

@@blackpenredpen in Canada 😂😂

[@Kradrice]

@@blackpenredpen damn I thought I saw you

[@blackpenredpen]

Kradrice oh then that must have been my twin! Jk

[@rhversity5965]

Can you do a video on the definition of an integral. Is it a summation?

[@bobdole69420]

The integral of a function is the area underneath the plot of the function. You can approximate an integral by splitting the area into lots of thin rectangles (obviously there will be some error here). You can describe this as the sum from i=a(the first part of the curve) to i=b(the second part of the curve) of f(x.i) (which is the height of the small strip of the curve, the length of the rectangle) * deltax (the distance between the strip of the curve, the width of the rectangle). When you take the limit as the rectangles get infinitely small, you get the exact answer - which is the integral.

[@angelmendez-rivera351]

Bob Dole ☭ Well... that's not quite true. The integral represents an area if and only if a number of restrictive conditions for the function hold. In general, though, it doesn't hold true.

[@AlBoulley]

Uhhh, search his existing videos for that already-answered question. Sheesh.

[@3862200]

I mean i always ask you questions but you never answered i wonder whyyy

[@user-ks7tu5qc9j]

Таким способом можно брать любой не берущийся интеграл.

[@meinteron6580]

e^(dx)

[@klausolekristiansen2960]

Don't let t and u bother t either.

[@saurabhayodhyase]

Hey sir i am from india My question is How to intigral of 1/e^tanx dx

[@bakrsalh6160]

Integral 1/sqrt(3+cosx)?????

[@floreskyle1]

integral of ln(arctanx) dx pls

[@justabunga1]

The answer is non-elementary.

[@cooldude_athar]

Ei(2)=?? Ei(1)=??

[@rurafs7934]

CPU 100%

[@ssdd9911]

dun let w bother u

[@jhoanantoni]

Gracias me ayudo mucho aunque no entiendo inglés:v

[@craigfowler7098]

All so obvious

[@williamadon9520]

Your video are great dude, not doubt about that but try to go slower if your purpose is teaching but it doesn't look so !!

[@hamnamubarak8247]

Love from Pakistan ❤

[@pervizzulfiqarov7487]

Like best video

[@dawandyudh7532]

??

[@ivangustavotoribiotorres5501]

Habla en español por fa

[@alexmak3004]

I just don't understand how can you do integration for years and still continue doing it. I find it boring.

[@thegr8fraz]

Don't let w double bother you!