I want all trig functions in one integral!

how calculus teachers create integrals for your test! I took me about an hour to craft this integral which features all the trigonometric functions, the natural logarithm function, the pi-th power, and a derivative for the denominator. This integral looks scary but the solution is actually very straightforward! Enjoy! #calculus #math #trigonometry #blackpenredpen

0:00 how calculus teachers create integrals for your test
6:30 check out Brilliant:
7:25 BOOM!

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Thank you all!

[JCCyC]

What the student really needs to learn here is not Calculus, but to control their own panic.

[kalculate]

i did not expect such a clean solution for this integral

[fortpile]

As a high school cal student, this is, so far, the most satisfying integral I've ever witnessed.

[MichaelZankel]

It’s always so funny to me when you add the sin(x)^2 + cos(x)^2 just to make it look harder but it’s just 1 😂 I love it lol

[Wmann]

“Everybody’s here”

I love that part.

[Generalth]

What a glorious result! I'm going to make this extra credit the first time I teach a calculus class!

[dinosauriaman]

I love how I'm in precalculus and I was still able to understand where you were able to go with this.

[MathZeimer]

There is nothing more satisfying than terms that perfectly cancel/match out, leaving a neat answer!

[gamerpedia]

Time to try it out!

sin²+cos²+tan²=1+tan²=sec²
(1-2ln(cos))^π

ln(csc)+ln(sec)+ln(cot)
=
= -ln(sin)
d/dx(-2ln(sin)) = -2 × cos/sin
= -2cot

So the integral becomes

sin×(1-2ln(cos))^π / -2cos
u = -ln(cos)
du = sin/cos dx

So
(2u+1)^π du
Where u = -ln(cos x)

Now another substitution
w = 2u+1
dw = 2du

So -¼×w^π dw

Integrates to
w^π+1 / (-4)(π+1)

w = 2u+1
u = -ln(cos x)
w = 1-2ln(cos x)

(1-2ln(cos x))^(π+1) / -4(π+1) + C

[johncraig]

This is the only guy that can make math fun. Keep up the amazing content!

[DownDance]

5:12 My favorite scene because we all know what he means 😂
This problem was actually quite easy for looking so difficult. The only things I didn't instantly see is the u-sub, because I didn't instantly know the derivative of sec(x)
Great video!

[tomasthemas]

I love how real the problem creation is. Start with a random smattering of operations and functions that look scary then adjust it until a solvable problem appears.

[MrWaltjam]

as a 50 year old I recently(2020-2022) went back and got my associates degrees after a debilitating health incident that left me reconstructing my base of knowledge. I also wanted to work my brain and ended up taking physics and calculus 1 and 2. At the community college level. I still have memory issues but these videos keep me atleast familiar with the concept of integration. I did well almost coming out with an A but for fractions of points in both.. graduated with a 3.92 gpa. the final Bs in calc I and II got my 4.0.. I can now help my highschool and middleschool children when they get stumped in their advanced math classes.
I love the depth to which you dive in explaining your process and the sloutions..Your humor is also not lost on me. Fun makes learning more bearable in difficult situations. I only wish my professor was so concerned with teaching the process. After the few example in the book, she assigned out homework. She wanted it the way she taught it but left us to "figure it out" no pun intended.. I would have drowned but for math is power, professor leanard and some of the others. I realized the value of not only knowing but proving the trig identites alittle too late but now I know. I may go back and finish the bachelors in meterology but am leaning toward mathematics just for fun. Keep up the good work, you are opening doors in the minds of many who were traumatized by higher maths as evidenced by the comments.. my only question is whether all the variations of blackpenredpen is actually you? there are several, its hard to tell these days.

[JoseIgnacioZapata]

You are so calm, smart and good natured I can't help but smile as I watch you solve these. Good work!

[nvapisces]

When i saw this, it sort of look solvable and now i see this being solved makes it so satisfying

[mineapple]

It’s always good when teacher give you things in a test you’ve never seen before, like cscx, secx and cotx

[huwdte]

This integral looks so daunting to complete, and watching it quickly simplify down is just really satisfying

[ricardomorales]

please make more "insane" looking integrals that are actually fairly easy! I loved the way you break down the solution for something so ridiculous looking :)

[RaiymbekZhasulanuly]

As a person learned basic integration only yesterday, this thing really is fascinating

[bylightIta]

I'm in high school and this year we finally did the indefinite integrals. Now I can comprehend some of this man’s videos. Nice solution