Battle 1, integral of cos(x^2) vs integral of cos(ln(x)), @1:00 Battle 2, integral of ln(1-x^2) vs integral of ln(1-e^x), @7:55 Battle 3, integral of x^(x/ln(x)) vs integral of x^x, @16:23 Battle 4, integral of x*sqrt(x^3+4) vs integral of x*sqrt(x^4+4), @19:29 Battle 5, integral of x/ln(x) vs integral of ln(x)/x, @32:25 Battle 6, integral of ln(ln(x)) vs integral of sqrt(x*sqrt(x)), @34:00 Battle 7, integral of sqrt(sin(x)) vs integral of sin(sqrt(x)), @36:13 Battle 8, integral of sqrt(tan(x)) vs integral of tan(sqrt(x)), @40:52 Battle 9, integral of tan^-1(x) vs integral of sin^-1(x)/cos^-1(x), @59:13 Battle 10, integral of 1/(1-x^2)^(2/3) vs integral of 1/(1-x^2)^(3/2), @1:04:23 file: docs.wixstatic.com/ugd/287ba5_3f60c34605f1494498f02a83c2e62b29.pdf

Another approach to the integral of ln(1-x^2) dx would be to factor the inside and then use the product rule of logarithms to get the integral of ln(1-x) + ln(1+x) dx. It's a bit easier to solve this way.

Solution to integral of sqrt(tan(x)): There's a blackpenredpen video on that + c

Just a real minor point of #4: you could also do a hyperbolic trig substitution instead, and you'd get a simple inverse hyperbolic sine term in the final answer instead of the natural logarithm. That natural logarithm is also convertible to the inverse hyperbolic sine.

I love your enthusiasm!

The way I like to think about the Integral of cos(x^2): with some clever substitutions and Euler's formula it can be shown that it can be written in terms of the integral of e^(x^2) and since that cannot be defined in terms of elementary functions, thus the integral of cos(x^2) cannot be

I want to know, how to prove that the integral of a function is not elementary, please tell

I love these videos! Encore, encore :-)

So nice to see a notification from bprp just after the first day of school :D

This is the best video You have made - of those I've seen. I was especially happy to know that ln(ln(x)) is a non-fundamental function. That question has been bothering me for years.

On the first one, it was obvious, because cos(ln x)=(x^i+x^-i)/2. Power rule, separate real and imaginary coefficients, and put it back to trig functions. Even if you're not going to use complex numbers, you can guess the right integral because cos is like an exponential and goes well with ln and poorly with x^2.

22:21 Euler's substitution sqrt(u^2+4)=t-u would be better idea here Last one third Euler substution (with roots) or integrating by parts also are good option

To integrate arcsin(x)/arccos(x) from x = -1 to x = t < 1, let x = cos(θ). Then dx = -sin(θ) dθ. The integrand is now -arcsin(cos(θ))·sin(θ)/θ. The bounds are from θ = π to θ = arccos(t). On the interval (0, π), which is the codomain and range of arccos(t), arcsin(cos(θ)) = π/2 - θ. Therefore, the integrand is -(π/2 - θ)·sin(θ)/θ. Factoring -1 will change the bounds to run from θ = arccos(t) to θ = π, with integrand (π/2 - θ)·sin(θ)/θ. By linearity, this gives the integrals of (π/2)·sin(θ)/θ and -sin(θ). The first integral is equal to (π/2)·(Si(π) - Si(arccos(t))), and the second is equal to cos(π) - cos(arccos(t)) = -(1 + t). Then the total integral is simply equal to [(π/2)·Si(π) - 1] - (t + Si[arccos(t)]). Call (π/2)·Si(π) - 1 = C, so the integral is simply C - t - Si(arccos(t)). Done! For the record, Si(x) is defined as the integral from s = 0 to s = x of sin(s)/s. We can extend the answer to other intervals, but this requires some caution, since arcsin(cos(θ)) = π/2 - θ is no longer true in other intervals.

One-hour long video but u definitely spent a lot more time than that! Your effort should be appreciated! And also the patreon list grows longer everytime 😁👍 PS it's 1am here in HK and yr thumbnail looks cool with some chill 😆

You are a great teacher

Take my love for this channel from Bangladesh.

7:55 wouldn't that be easier to just factor 1-x^2 as (1-x)(1+x) and then use the log propertry to split the ln of the product?

15:05 In the last two terms of that answer (before the +C) it was not necessary to use absolute value around the ln input. Respond to this comment if you can figure out why!

one take, with some cuts. i dig it 😁

19:57 you can do both u-sub and trig-sub at the same time by letting x^2=2tan(theta) ;) then, xdx is nicely equal to sec^2 and the rest is just the usual

Hey BPRP , can you make a video about Group Theory ?

What font did you use in your document? Do you use LaTeX package or?

Brilliant sir

I came here after the rap battle in 8 Miles😂... I am ready for the battle!!!

Would it actually be faster to integrate cos(ln(x)) by using the complex definition of the cosine? You would then need to integrate (x^i+x^-i)/2, which is just a matter of integrating polinomials.

On question number (8). Suppose you let integral equal to Q, then square both sides and integrate twice then take the sqr,, can it work?

who else got a smile on the face at 16:15 because you have watched an old bprp video?

Hey Im a Calculus amateur. Just wondering what method did bprp used at 38:50. Thx in advance!

Integration of e^-xx from +inf To -inf with pler co-ordinates

which integrals are intermediate and high school?

can show hyperbolic functions more love or not?

Lets see those special functions!

The second one was a bit over the top, ln(1-x^2)=ln((1-x)(1+x))=ln(1-x)+ln(1+x)

Did You make already any video with non-elementary integrals like eliptic ones?

Number 9 is a pretty straightforward battle, once you know the formula for antiderivatives of inverse functions. As long as a function has an elementary antiderivative, its inverse has an antiderivative of the form, xf^-1(x) - F(f^-1(x)). Once you know tan(x) has antiderivative ln|sec(x)| + C, you just plug tan^-1(x) into the formula and do some trig identities on sec(tan^-1(x)) to get the same result.

Isn't it instead of using partial fractions, Can we not have xln (1-x²) -2x + tanh-¹ (x) +c ? As the answer?

can you know if the integration is possible or not just by looking at it ? , and if yes how do you know ?

Best videos sir for maths

Hey bprp, what font do you use in your files and thumbnails?

Sir ,What is the integral of ∫(1-x^2)^n dx

Hi BPRP, and thank you for the videos :D I guess this comment will go unnoticed, but if I never ask, I'll never know :) Why are half of these functions impossible to integrate? You just mention as a fact that it's impossible but never why. I'm not great at integration, so I don't understand _why_

It will be a great pleasure to me, if you explain how to separate elementary from nonelementary ones. Does such formular exist?

Correct me if I am wrong, but at 8:50, you can factor 1-x^2 and use rule of log to expand it into 2 terms?

Sir please make a video on ramanujan formula on finding value of pi

15:15 „Integrale für Euch“ 😂 Grüße an alle Deutsche 🇩🇪🙌🏽

You probably made that future video already, but it is interesting to point out that the most obvious attempt to antidifferentiate arcsin(x)/arccos(x) with respect to x results in the sine integral: A basic trigonometric identity has arcsin(x)=π/2−arccos(x), from which the integrand becomes ½π/arccos(x)−1; then the substitution x=cos(y) with dx=−sin(y)dy results in the sine integral. That is, ∫arcsin(x)/arccos(x) dx = -x−½π∫sin(y)/y dy = −x−½πSi(arccos(x))+C.

LETS GO!

salut monsieur svp j'aimerais avoir un pdf des 100 integrale ou un pdf d'çntegrale pour licence de mathematiues svp

How to do that (long division)?

integrating ln(cos x) would be an interesting one

√tanx i love this integral same as 1/(x^6+1)

Can you solve it Int. (x-2)/[(x-2)^2(x+3)^7]^1/3

Sir, why don't you make a video about proving that the ramanujan formula

What about x^dx? Can u do ir pls?

Thumbnails are getting stronger

integrating arcsinx/arccosx is actually doable;much easier to do than the other ones mentioned as undoable previously. its just a bit of subs and ibp and using the Si function.

Now solve the special function ones!

Battle 2: Don't use partial fraction! Use ln(ab) = lna + lnb rule first, much more simple!

Big salutation from Algeria thank you Allah blesses you

How we can know what is elementary and what is not?

In India we have National Teachers' Day on 5th Sept. So, Happy Teachers' Day to BPRP and all other teachers in advance.

Why IS integral of tan (sqrt x ) impossible to solve I genuinely don't understand

For no. 8, can't we split 1/(t^2-2) into partial fractions and use ln? It is much friendlier than coth. Also, why coth instead of tanh?

Please make a collaboration video with 3blue1brown together

hello brother. I get a different answer for number 2 intergral ln(1-x^2)dx instead of 1-x i get x-1 and 1+x is same as x+1

Number 8 was crazy

Isn’t it easier on the 2nd one to change it from ln(1-x^2) to ln((1-x)(1+x))=ln(1-x) + ln(1+x) and integrate like that?

For #9, the ln part turned out to be ln|cos(arctan(x))|, anyone else have this??

May the chenlu be with your integrals.

for number 2, isn’t the int of 2/1-x^2 just 2arccot(x)?

Hi, cos(X square) is a function . Geogebra gives a result, if you integrate ( calculate the area) between 2 points Why we can say that this integral does not have a result.thank you For your reply

the ad I had for this just said "Find your Steve" 😱😱😱

I like your microphone

Wait... 1 hour 😯💚

Here's another way to write the answer to question 2, xln(1-x^2)-2x+2tanh^-1(x)+C

How do you make your thumbnail🙏😊

12:30 you could just directly integrate it to 2tanh^-1(x). instead of partial fractions.

Good video, can you please help me with this integral .. X*Sec(X)

Second round: integral 1 / (1-x^2) = arctanh x + C

BPRP how to find range of Sinx-√3.cosx+1

How did he found out that we can't do the other one?

Can you teach us group theory?

Why cos(x^2)is impossible?? Can you try to integrate it???

Could you solve this integral? Integral of (secx)^(3/2). I wish you did it. Thanks for giving a lot of support

11:22-11:25 the integral of the thing you are saying needs partial fractions doesn't, actually, because the answer is clearly inverse hyperbolic tangent (Argthx/Argtanhx)

I never forget the chendu😆

Question 3, the absolute troll

What do you mean by elementary?

Lol, I speak Irish but I don't know if that helps in the slightest

On 14 September it is teacher's day in India . Please make a excellent special video on the day.

Yall I was just vibing to the Doraemon theme song in the beginning.

Only between you and me!😁

But how do we know it's impossible to integrate cos(x^2)?

bprp: *showing 8 integral battle* me: ...here we go again

m8 im in high school learning quadratics XD could u do a video where u explain calculus and why it works sorry i just kinda don't get what ur doing and just don't get calculus - but i still sub

PLeaSe CheCk tHe No.8 OnE wiTh DiFFerEnciAtiON

Do you think that Isaac newton would have been able to derive all of these integral solutions back in his day

Battle 8 is the best integral....

26:25 100 Integrals #61.

hi int(sinx^cosx,x)=? thanks