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So, I finally have a chance to show you how to solve a definite integral by taking the advantage of its symmetry. Don’t just use brute force and integrate all the time (although it might be more fun).

I got this problem from: https://brilliant.org/problems/jee-mains-2015-1530/ , and there are plenty more!!!

integral of log(x)/(log(x)+log(6-x)) from 2 to 4,

integral of sqrt(cot(x))/(sqrt(cot(x))+sqrt(tan(x))),

JEE main 2015 integral,

integral by symmetry,

integral properties,

blackpenredpen,

math for fun

source

It is worthwhile to teach students to first LOOK at a plot of the integrand of a definite integral. See what you're up against. Things like symmetry stand out visually. And so does the fact that this function is quite close to linear over the interval. We happen to need the former and not the latter in this particular case, but irl it may well be the other way around.

Can i spoil it?

10:42 YAAAAAAAAAAYYYYYYYYYYYYYYYYYY

Suggestion: record what you're doing in the phone, then added to the final video. I think it'll be better than showing us the screen of your phone. Good video, btw.

8:48 Together, u and I can solve this integral

wow, this integrale was pretty cool

I remember a video of you when you used the complex integral of x^i in order to evaluate the real integral of cos(ln(x))

You introduced complexes in a non complex problem to make your life easier.

I'm looking forward other videos where you'll use complex to easily solve real equations!!

1 isn't it?

Can you make a video on calculating integrals by symmetry? I am still not sure

@blackpenredpen I wonder if you can find the Integral of x^x from 0 to pi.

Edit: Bonus: Integral of same function from 0 to e, and more hilarious Integral of same function from 0 to i?

Comedic Edit: I'm a fool, I should have seen you video on Integral of x^x before saying this. 🙁

Hey, just curious. Do you do these equations beforehand or do you do them for the first time in these videos?

Does this mean that in the interval log x equals log (6-x)?

we can use integral a to b f(x)dx

=integral a to b f(a+b-x)

After second step….

This uses a basic symmetry property:

int_a^b f(x) dx = int_a^b f(a + b – x) dx

Thus, our integral

I = int_2^4 (log x)/(logx + log(6 – x)) dx

becomes

I = int_2^4 (log(6 – x))/(log(6 – x) + log(x)) dx

Then,

2I = int_2^4 dx = 4 – 2 = 2

I = 1

Spoiler: The next video also uses this property 🙂

For the integral at the end of the video, I think it's pi/4 (???)

I simplified it to just cos and sin and did a weierstrass substitution followed by partial fraction decomposition and was able to integrate, and at the end it was just arctan(1).

Probably did it more complicated than it had to be but it seemed to work lol

If anyone could check me, I'd be grateful

For the question at the end, I got infinity since we're equivalently integrating csc(x) from 0 to pi/2. Is that correct?

I'm preparing for the same exam lol.

7:51 What'd you just call me?

As a student of Calculus, I m thankful to you sir for solving this problem. It would be really nice of you if you do more of these videos based on IIT-JEE Questions because these are quite challenging plus you will also get a boost in your views becuase IIT JEE is a big thing in India…!

I can feel his happiness 1:01 hahaha

Wait so actually all types of logs will have the same result in it if you changed the logs to lns it will have the same answer that's so cool

If you let u=3-x, I think things cancel out easier.

integral argument becomes:

u=1 to u=-1

log(3-u) / [ log(3-u) + log(6-3-u) ] * (-du)

this can factor to

u=-1 to u=1

[ log(3-u) / log(3-u) ] * [ 1/ (1 + 1) ] du

=>

1 * (1/2) du

then

u/2, from -1 to 1

=>

(1)/2 – (-1)/2 = 1

Please also do questions from iit jee "advanced" as well which is considered to be tougher than iit "mains"

try this upper limit 3½ ( mixed fraction), lower limit ½ , f(x)= {½(|x-3|+|1-x|-4)} integrate w.r.t x. |.|=modulus function {.}=fractional parts function

How can we let u=6-x, then rewrite the formula, and then just swap u for x, without paying attention to the definition we have given u? Can we just let go of u=6-x later in the equation do that 6-u==6-x? Seems odd to me

Very good sir

I am class 12 student in India

And found it very interesting!!!!??????

like your "stay in shape" comment. : )

Thank you!! You accidentally helped with my homework! <3

hey……i did this question few days ago……i am preparing for jee main and advanced 2018…..plz upload more questions like this from Jee Advanced …..!!

wtf who asks for views!!

Hey man, I LOVE your videos, especially this one haha. One thing I dont quite get though is why you are able to just substitute x for u at 7:20 . I mean u = 6-x isnt it? So why can you say that u = x and just get the integral into the x world. It'd be great if someone helped me out here, its a beautiful integral and it'd be even more beautiful if i understood that part 🙂

I love watching you go through these problems

You Just Dont Know Integeral PROPERTIES bHAI !!!

10:42

YAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAY!!!!

Dont solve JEE mains questions do JEE advanced 😀

Love and blessings from India!!

this notation something I dont like about my maths book (ncert and other indian class 12) book that they use log.. for natural logs. It can quite confuse anybody unknown. and may lead to wrong answers due to misconception.

why not use standard ln for log base e and log for log base 10.

DON'T LET u BOTHER YOU

You are about halfway there

Imagine solving them in less than 2 min

Please solve more JEE mains and advanced questions on integration and other topics.