What can I look for? 1:00 > Pythagorean theorem works. The function can be one side. [3:30] The idea of trigonometric substitution 7:00 Example 11:00 - x = tan(theta), dx/dtheta = tan (theta) > dx = sec^2(theta) dtheta integral of sec^3(x): th-cam.com/video/6XlSP58u-is/w-d-xo.html > convert the problem back into terms of x The skinny on how to set up trig sub 18:30 (good conceptual discussion) Examples 36:00 Another example 54:00 >> look for a square root in your problem If a definite integral problem, change your bounds to avoid having to change back into x terms, which requires revisiting the triangle Example 1:11:00 > trig identities 1:24:00 If it doesn't fit, change to sines and cosines 1:26:00 Consider the function as it stands now. What method works best? Look at all the techniques > In this case, don't use more identities... but appreciate that U-sub works here. Reminder for U:Sub: You want to turn the function into something on the integral table. picking cos as u would leave a sin behind, but differentiating sin would cancel out the cos i.e. replace it with du Final example 1:32:00 Process: Is it in the table? U-Substitution? IBP? Trig-Sub, does it have a square root? > if it doesn't fit 'something squared minus something squared, change that. In this example, complete the square [1:35:20]. This was the idea because the stuff under the square root resembles a quadratic/polynomial 1:40:30 substitute to get rid of 'x+1' 1:52:00 sec^4(x) is an even power >> use identities [1 + tan^2(x)] to cancel out sec^2 via derivative and substitution rules > note: half angle identities are power reduction formulas. > video for integral of cos^4(x): th-cam.com/video/NSAwZNzjex4/w-d-xo.html Substituting back 1:56:00
I swear you're one of the most professional and experienced professors I can ever find to teach math. You simply love what you're doing and that's what makes you an excellent professor. Keep going big man 💪🏻
Cant believe I paid $1200 UC tuition for my 5 week online calc 2 class when I only use your videos to learn the material. THANK YOU for my first A+ in a STEM class, I seriously couldn't have done it without you. I subbed to your patreon, hopefully one day I can do more but I'm eternally grateful.
+mrburns805 same, I go to a highly reputable engineering college but guess what, I learn most of my mathematics and science from youtube which is kinda sad b/c college is expensive :(
Your professors don't suck, it's just that they don't know how to speak dumb. They went into teaching math because of how well they get it, not to mention how well they get it after they've gone way beyond undergrad with it. They don't know how to speak dumb and they take their knowledge of it for granted. I don't blame them, but this guy is just gifted at seeing it how we see it.
You are truly an amazing professor. I wish my Calc 2 teacher taught as well and understandable as you. All she does is practice problems and no explanation of concepts, formulas, or theorems. You are my new Calc 2 professor, keep up the amazing teaching!!!!
I have always had to learn the trig subs off by heart. Man, for the first time I really understand trig subs thanks to your method of UNDERSTANDING WHAT WE ARE DOING. A million thanks to you Professor Leonard, you are THE BEST.
No wonder I'm failing calc 2. My teacher sucks at explaining stuff unlike this guy. Hopefully I can binge on these videos and catch up in time to pass!
These videos are a god-sent. You have no idea how much of a hero you are to so many students!! I spent hours on practice problems for this stuff but once you explained it everything was so clear and I could EASILY do problems. You make me interested in calculus and math for the first time in years. You and your class are so sweet and funny which makes every video that much better!
Professor Leonard you probably know how brilliant you are. I am in college third year and so far I have seen only couple teachers who explains clearly. I am taking calculus 2 but you are my teacher.
Man it is so nice to understand what is going on in Calc II, my teacher does not explain anything and it just goes right over my head. These videos have saved me
"Don't just fall into the pattern of looking at the formulas. I HATE THAT!" 57:43 Preach it! I love how righteous he gets about being a lazy math student. The strength o f Leonard's methods is that they never rely on memorization of formulas. Too many students fall into the trap of relying on memory instead of employing their understanding of concepts.
Thank you so much for these! They stick so well in the memory when you get where these formulas come from... And it gives you a strong fallback solution in case you forget them. Just draw the triangle and it hits you.
Hi Prof. Leonard, I've been using your videos to review some trig sub and am doing the examples before you solve them. Thank you so much for all the videos you make, it's very comforting to have you guiding me through all this. Hope you're doing well, can't wait to see your new videos!
It's been a few years since I took calc 2. But this is the first time I feel like I fully understand this as a concept. I was able to get through the class perfectly fine just memorizing the formulas but after I completed the class I often found I had to keep looking up the substitutions every time I had to use this method again. I had a friend taking calculus at the same time that told me there was a better way where you can derive the formula's each time by starting with the right triangle. I always liked this approach and wanted to use it but I never fully understood it because I didn't know what to pick for my sides. I didn't realize that there was some ambiguity to it and you could draw two possible right triangles for each substitution. And nobody told me that the other substitutions that we never use, i.e. with cotangent, cosine and cosecant work.This explanation finally made it click for me. It's not that we have to draw the triangle in the way you draw them, you just pick the triangle that allows us to integrate something more convenient. For the first integral we could have just as easily integrated the csc^3 instead of sec^3 to get the answer, which I verified just now. But integrating secant is a little less cumbersome than integrating cosecant.
I watched this lecture and felt like I learnt a tonne. I give another TH-cam teacher a go on the same topic and the only reason I could follow along was because of this video. The other video was just a plug and chug formula video there’s no way you could learn and remember much from it. On the other hand professor Leonards lecture was unbelievable, a very good intro to the theory and why it works followed by fully worked solutions with explanation of each step, perfect way to learn 👍
I wish he was my professor, my professor literally just does examples on the board super fast while facing the board and doesn't answer questions - I am so happy I found you!
you tech calculus perfectly, i love that you get your students engaged and make sure they know their stuuf and omg you actually explain how things work unlike most of my college profs ugh also... u look like a dad
OOOH MY GOD I don't know how to thank you Professor. My academic coach Stephen recommended this video to me. thanks to him, I feel better about trig sub now.
It’s crazy to me how after I graduate I have to pay back loans for a teacher that doesn’t even care to teach well. Most Calculus professors are just graduate students forced to teach to do their research and grad work, and many of those don’t spend the time necessary to make sure student are getting the absolute most out of their classes. You really can judge how well a teacher is doing based on how many hours students spend re-learning the concept outside of class. Professor Leonard is one of the best calculus teachers I have ever seen, he intuitively breaks down the mechanics of concepts, engages the class routinely, and never skips 3 or 4 steps while teaching, making sure to cover every single logical step that takes place even if it’s simple 3rd grade arithmetic. We need to find a way to get our student loans to filter to people like Professor Leonard instead of our professors. Professors get the same pay regardless of how well they teach, but if their students are consistently resorting to relearning each concept on Khan Academy, Professor Leonard, or Paul’s Notes, then that money should be going to those people, not the school. If schools know they can get more money with better teachers, then that could be a step in the right direction. While I love Professor Leonard, the goal of education is to make sure Professor Leonard’s videos are not necessary.
Professor Leonard, thank you for an incredible and lengthy video/lecture on Integrals by Trigonometric Substitution in Calculus Two. Trigonometric Substitution is not difficult; however a basic understanding of old fashion Trigonometry is needed. I also encountered Trigonometric Integrals in Engineering Physics and Electromagnetics. This is an error free video on TH-cam.
I recommended your lectures to my classmates, that was awesome i had a hard time tryin to figure out what my prof was doing when he gave us this subject.
I was beginning to think this guy was a T-1000 Terminator sent from the future and disguised as a math teacher until 1:15:30 when he dropped that pen. I guess he IS human. He's just THAT good.
Prof. Leonard... Thanks for being a seriously excellent teacher. I'm fairly certain you could teach my dog calculus. I'd be failing Calc II if it wasn't for you.
One point of consideration is to go ahead and transform the integrand immediately from the quadratic radical to the trigonometic functions from the triangle rather than going back to the integrand and then using the Pythagorean identity to convert to the substituted rational trigonometric function. For example, from the triangle you can directly convert the radical expression to 4 tan v = (x^2-2^2)^1/2. With that you can take your 2 sec v = x and then derive the differential 2secv tanv dv = dx. From here, you are ready to roll with the direct integration of the substituted integrand. One other point to emphasize is that trigonometric substitution is only appropriate for a quadratic radicand. If the radicand is linear, then an nth root substitution is very simple. For example if the integral were the third root of 9x + 1, then let u^3 = 9x+1, which yields x = (u^3 - 1)/9. So, dx = u^3/3 du. The point is the whole crux of solving integrals is knowing, which method is appropriate and most efficient, and in this case the degree of the integrand is the decisive criterion to know whether to use an nth root or trigonometric substitution. Overall, your explanations are really great. I have seen so many do such a poor job of explaining trigonometric substitution. Using the right triangle is absolutely the best way to work these problems rather than transitorily memorizing some equations for the appropriate substitutions. The whole concept is rather straight forward as far as the substitution goes. It can become quite involved once you start actually integrating with all the exponentiated trigonometric functions. In any event, great video and great techniques. Good on you! Great job. You are a credit to mathematical instructors everywhere.
Hi! I'm using your Calc 2 playlist to help me prepare for my Calc BC test (it's not offered at school, unfortunately) and your videos are a lifesaver!! Thank you so much! =D
always make sure o like every video i watch!! you the best prof! my teacher should take a look at you channel to learn how to teach!! thank you, you saved my life
Thank you Sir ! i rally appreciate your great job. the way you explain it is very simple and understandable! i am sure it is helping a lot of people like me! Thanks again
Suprised that so many students are learning difficult integration. You'd think that not many people in the work force would need to understand skills like this,
At last. I can understand what was 'taught' to my class, leaving us wondering what was going on? Calculus is not too difficult. I wish I had known about these videos then. Wait, that 20 years ago. Professor Leonard would not be where he is today! Oh well . . .
I have a question: at the beginning of the video you said that integration by parts is not possible because "we don't have another part" - however; we could have used "1" as the other part - sometimes we do this with integrals like the integral of ln(x). Not that we should use integration by parts, since taking the derivative or integral of that would make the integral more complicated, or that I don't like your videos (I personally LOVE them), I just thought that it would be an important clarification to make :)
I looked up this video because I don't know which side the x should be at. Now I know but I still don't get why we have to pick the x on that specific side?
No homo. Such a great and good looking professor. Funny, kind, patience, good looking. What a great professor you are sir. Thank you for the knowledge.
Dear sir, is it possible to make,for instance, a u-substitution where the differential (du) is located in the denominator? Thank you very much for all the material you posted online.
you should call your channel Calc Kent.
saving the world one differentiation at a time
I always thought the S looked like an integral symbol
What can I look for? 1:00
> Pythagorean theorem works. The function can be one side. [3:30]
The idea of trigonometric substitution 7:00
Example 11:00 - x = tan(theta), dx/dtheta = tan (theta) > dx = sec^2(theta) dtheta
integral of sec^3(x): th-cam.com/video/6XlSP58u-is/w-d-xo.html
> convert the problem back into terms of x
The skinny on how to set up trig sub 18:30 (good conceptual discussion)
Examples 36:00
Another example 54:00 >> look for a square root in your problem
If a definite integral problem, change your bounds to avoid having to change back into x terms, which requires revisiting the triangle
Example 1:11:00
> trig identities 1:24:00 If it doesn't fit, change to sines and cosines
1:26:00 Consider the function as it stands now. What method works best? Look at all the techniques
> In this case, don't use more identities... but appreciate that U-sub works here. Reminder for U:Sub: You want to turn the function into something on the integral table. picking cos as u would leave a sin behind, but differentiating sin would cancel out the cos i.e. replace it with du
Final example 1:32:00
Process: Is it in the table? U-Substitution? IBP? Trig-Sub, does it have a square root?
> if it doesn't fit 'something squared minus something squared, change that. In this example, complete the square [1:35:20]. This was the idea because the stuff under the square root resembles a quadratic/polynomial
1:40:30 substitute to get rid of 'x+1'
1:52:00 sec^4(x) is an even power >> use identities [1 + tan^2(x)] to cancel out sec^2 via derivative and substitution rules
> note: half angle identities are power reduction formulas.
> video for integral of cos^4(x): th-cam.com/video/NSAwZNzjex4/w-d-xo.html
Substituting back 1:56:00
th-cam.com/video/y_XwQkchwrE/w-d-xo.html
Great work.
thank you so much❤
A hero
Best trig substitution lecture EVER!!! Thank you Professor Leonard!
I swear you're one of the most professional and experienced professors I can ever find to teach math. You simply love what you're doing and that's what makes you an excellent professor. Keep going big man 💪🏻
that looks just like his arm
Cant believe I paid $1200 UC tuition for my 5 week online calc 2 class when I only use your videos to learn the material. THANK YOU for my first A+ in a STEM class, I seriously couldn't have done it without you. I subbed to your patreon, hopefully one day I can do more but I'm eternally grateful.
Are you like a math professor now
I go to college to take calculus 2, but you are my teacher, Professor Leonard. Thank you so much
+mrburns805 same, I go to a highly reputable engineering college but guess what, I learn most of my mathematics and science from youtube which is kinda sad b/c college is expensive :(
+bell hill
You should have gone to a community and then transferred.
+bell hill exactly me i could not learn anything from college
SKYNET #2025 That's exactly what I am doing!
Yep I can’t even understand my teacher lol
Let's all sign a petition to bring back Prof. Leonard!
He's working with Sal Khan to run for president in 2028
This is some good stuff right here.
holy fucken shit do not skip the first part of this lecture! that is the KEY the GOD! my professor didn't say it.
Some of the best calculus videos out there. Very thorough. Thank you.
how is it that this guy is so cool. hes way better than my prof teaching this.
I know right all my calc professors have sucked i've been using his lectures since beginning of calc 1 almost exclusively.
Your professors don't suck, it's just that they don't know how to speak dumb. They went into teaching math because of how well they get it, not to mention how well they get it after they've gone way beyond undergrad with it. They don't know how to speak dumb and they take their knowledge of it for granted. I don't blame them, but this guy is just gifted at seeing it how we see it.
You are truly an amazing professor. I wish my Calc 2 teacher taught as well and understandable as you. All she does is practice problems and no explanation of concepts, formulas, or theorems. You are my new Calc 2 professor, keep up the amazing teaching!!!!
I have always had to learn the trig subs off by heart. Man, for the first time I really understand trig subs thanks to your method of UNDERSTANDING WHAT WE ARE DOING. A million thanks to you Professor Leonard, you are THE BEST.
No wonder I'm failing calc 2. My teacher sucks at explaining stuff unlike this guy. Hopefully I can binge on these videos and catch up in time to pass!
+aliphim I'm in the same boat, buddy. Let's destroy this shit!
How did it go for y'all?
@@nuetro1709 They ded.
we usher in the newest generation of believers @@DAT-OFFICIAL
aliphim same
These videos are a god-sent. You have no idea how much of a hero you are to so many students!! I spent hours on practice problems for this stuff but once you explained it everything was so clear and I could EASILY do problems. You make me interested in calculus and math for the first time in years. You and your class are so sweet and funny which makes every video that much better!
Outstanding sir.Your methodology towards the students is up to the mark.
Like from Paistan⭐⭐
He makes Calculus 1 & Calculus 2 so much easier. Thanks Professor Leonard!
Professor Leonard you probably know how brilliant you are. I am in college third year and so far I have seen only couple teachers who explains clearly. I am taking calculus 2 but you are my teacher.
Man it is so nice to understand what is going on in Calc II, my teacher does not explain anything and it just goes right over my head. These videos have saved me
"Don't just fall into the pattern of looking at the formulas. I HATE THAT!" 57:43
Preach it! I love how righteous he gets about being a lazy math student. The strength o f Leonard's methods is that they never rely on memorization of formulas. Too many students fall into the trap of relying on memory instead of employing their understanding of concepts.
Thank you so much for these! They stick so well in the memory when you get where these formulas come from... And it gives you a strong fallback solution in case you forget them. Just draw the triangle and it hits you.
No one comes close to your "can do"attitude!
Is this a regular college class? I swear we only spent 30 minutes in my class with this topic.
You made me pass my finals and now I’m an engineer thanx to you ! Always keep you in my prayers my man 🙏
what did you end up doing with the degree? you making any money?
thank you for starting from the most basic understanding of a problem, it really helps.
Hi Prof. Leonard, I've been using your videos to review some trig sub and am doing the examples before you solve them. Thank you so much for all the videos you make, it's very comforting to have you guiding me through all this.
Hope you're doing well, can't wait to see your new videos!
Why couldn't my teacher be more like you! Thanks so much for the videos, will save my butt this semester.
It's been a few years since I took calc 2. But this is the first time I feel like I fully understand this as a concept. I was able to get through the class perfectly fine just memorizing the formulas but after I completed the class I often found I had to keep looking up the substitutions every time I had to use this method again. I had a friend taking calculus at the same time that told me there was a better way where you can derive the formula's each time by starting with the right triangle. I always liked this approach and wanted to use it but I never fully understood it because I didn't know what to pick for my sides. I didn't realize that there was some ambiguity to it and you could draw two possible right triangles for each substitution. And nobody told me that the other substitutions that we never use, i.e. with cotangent, cosine and cosecant work.This explanation finally made it click for me. It's not that we have to draw the triangle in the way you draw them, you just pick the triangle that allows us to integrate something more convenient. For the first integral we could have just as easily integrated the csc^3 instead of sec^3 to get the answer, which I verified just now. But integrating secant is a little less cumbersome than integrating cosecant.
I watched this lecture and felt like I learnt a tonne. I give another TH-cam teacher a go on the same topic and the only reason I could follow along was because of this video. The other video was just a plug and chug formula video there’s no way you could learn and remember much from it. On the other hand professor Leonards lecture was unbelievable, a very good intro to the theory and why it works followed by fully worked solutions with explanation of each step, perfect way to learn 👍
I wish he was my professor, my professor literally just does examples on the board super fast while facing the board and doesn't answer questions - I am so happy I found you!
you tech calculus perfectly, i love that you get your students engaged and make sure they know their stuuf and omg you actually explain how things work unlike most of my college profs ugh also... u look like a dad
You're an awesome teacher. You explained much better then my Calculus II teacher did. I'll be recommending your videos to my classmates.
Thank you with awesome two years with your videos Professor Leonard, I pass Calculus 1,2,3 THANKS TO YOU!!! FREEEEEEE
Thank God for this man. Professors/textbooks don’t go in depth or are as clear as you.
Anyone notice that this man built like a GOD
Thank you Professor, you have made what was daunting and confusing, in to engaging and understandable!
Lots Of love From China.Thank you Sir❤.Just because Of your teaching method I feel like my base is improving day by day.
thank you very much you are very good teacher i hope to have a teacher like you
اشكرك انت استاذ رائع اتمنى ان اقابل استاذ عظيم مثلك
OOOH MY GOD I don't know how to thank you Professor. My academic coach Stephen recommended this video to me. thanks to him, I feel better about trig sub now.
It’s crazy to me how after I graduate I have to pay back loans for a teacher that doesn’t even care to teach well. Most Calculus professors are just graduate students forced to teach to do their research and grad work, and many of those don’t spend the time necessary to make sure student are getting the absolute most out of their classes. You really can judge how well a teacher is doing based on how many hours students spend re-learning the concept outside of class. Professor Leonard is one of the best calculus teachers I have ever seen, he intuitively breaks down the mechanics of concepts, engages the class routinely, and never skips 3 or 4 steps while teaching, making sure to cover every single logical step that takes place even if it’s simple 3rd grade arithmetic.
We need to find a way to get our student loans to filter to people like Professor Leonard instead of our professors. Professors get the same pay regardless of how well they teach, but if their students are consistently resorting to relearning each concept on Khan Academy, Professor Leonard, or Paul’s Notes, then that money should be going to those people, not the school. If schools know they can get more money with better teachers, then that could be a step in the right direction. While I love Professor Leonard, the goal of education is to make sure Professor Leonard’s videos are not necessary.
Professor Leonard, thank you for an incredible and lengthy video/lecture on Integrals by Trigonometric Substitution in Calculus Two. Trigonometric Substitution is not difficult; however a basic understanding of old fashion Trigonometry is needed. I also encountered Trigonometric Integrals in Engineering Physics and Electromagnetics. This is an error free video on TH-cam.
I NEED THIS MAN AS MY PROFESSOR, NJIT NEEDS YOU
I recommended your lectures to my classmates, that was awesome i had a hard time tryin to figure out what my prof was doing when he gave us this subject.
God will bless you my math professor. I am very happy with this videos so much because they really solve my problems. my problems are settle
It's so amazing how he makes the students have fun while learning something as hard as this
I was beginning to think this guy was a T-1000 Terminator sent from the future and disguised as a math teacher until 1:15:30 when he dropped that pen. I guess he IS human. He's just THAT good.
Prof. Leonard... Thanks for being a seriously excellent teacher. I'm fairly certain you could teach my dog calculus. I'd be failing Calc II if it wasn't for you.
One point of consideration is to go ahead and transform the integrand immediately from the quadratic radical to the trigonometic functions from the triangle rather than going back to the integrand and then using the Pythagorean identity to convert to the substituted rational trigonometric function. For example, from the triangle you can directly convert the radical expression to 4 tan v = (x^2-2^2)^1/2. With that you can take your 2 sec v = x and then derive the differential 2secv tanv dv = dx. From here, you are ready to roll with the direct integration of the substituted integrand. One other point to emphasize is that trigonometric substitution is only appropriate for a quadratic radicand. If the radicand is linear, then an nth root substitution is very simple. For example if the integral were the third root of 9x + 1, then let u^3 = 9x+1, which yields x = (u^3 - 1)/9. So, dx = u^3/3 du. The point is the whole crux of solving integrals is knowing, which method is appropriate and most efficient, and in this case the degree of the integrand is the decisive criterion to know whether to use an nth root or trigonometric substitution.
Overall, your explanations are really great. I have seen so many do such a poor job of explaining trigonometric substitution. Using the right triangle is absolutely the best way to work these problems rather than transitorily memorizing some equations for the appropriate substitutions.
The whole concept is rather straight forward as far as the substitution goes. It can become quite involved once you start actually integrating with all the exponentiated trigonometric functions. In any event, great video and great techniques. Good on you! Great job. You are a credit to mathematical instructors everywhere.
Am rally inpress about everything, because such explianation/lecture for those luck students
Muy buena clase , saludos desde ¡Colombia!
th-cam.com/video/y_XwQkchwrE/w-d-xo.html
Hi! I'm using your Calc 2 playlist to help me prepare for my Calc BC test (it's not offered at school, unfortunately) and your videos are a lifesaver!! Thank you so much! =D
Haha im doing the same thing, hopefully I get to start college at calculus III (:
You have a good heart Professor Leonard !
Wow! Although I followed everything he said I wouldn’t want an integral like that in a examination question!
always make sure o like every video i watch!! you the best prof!
my teacher should take a look at you channel to learn how to teach!!
thank you, you saved my life
Thank you Sir ! i rally appreciate your great job. the way you explain it is very simple and understandable! i am sure it is helping a lot of people like me! Thanks again
amazing you are an awesome teacher I was so lost in class but now it makes sense
Wonderful trig substitution professor Leonard Thanks alot
This man is incredible.
You...are....amazing! thought this lesson was extremely hard until I watched this video!
please don't stop man. you saved my tears
best videos for calculus ever❤️
I LOVE THIS GUY! YOU'RE AWESOME !!! KEEP UP THE GREAT WORK MAN!
Excelent lecture professor!
Suprised that so many students are learning difficult integration. You'd think that not many people in the work force would need to understand skills like this,
It is still like 0.01% of people.
am really enjoying this Tutorial Sir
That was a nice lecture ! I enjoyed it :)
At last. I can understand what was 'taught' to my class, leaving us wondering what was going on? Calculus is not too difficult. I wish I had known about these videos then. Wait, that 20 years ago. Professor Leonard would not be where he is today! Oh well . . .
thanks professor, this is very high quality material
You just spared me weeks of confusion. thank you!!
This guy is a math legend!!
this video just might save me for my midterm
MIDTERM 2 in 1 hour and i am learning this nowww, hope it'll help mee
I love him. thanks for saving my calc grade
This teacher IS ballin
Professor Make a Pateron Alot of people are more than willing to donate to you ! BEST
I got 10/40 on my midterm , i need at least 40 marks on the final to pass 😂😂😂😂😂 am i cooked !!?
the baby gap T crushed it again! My man! Donald Trump gotta make you head of the education department, keep killin it brotha.
Dude, you're the best!
Professor Leonard, you are the Richard Feynman of mathematics!
Hi professor brilliant as always,would you set a video about the multivariable after completing calculus 2 ?
''best in the world at what he does''
We barely covered this in class :(
Thanks :D
th-cam.com/video/y_XwQkchwrE/w-d-xo.html
great visualizer, fantastic teacher, and huge biceps... What else could u ask for in a calc professor
AGREED!!! XD PASSED CALC SOOO EASY WITH THIS BRO HAHAHA LOLOLOLOL xD xD xD
But for real. Prof Leonard is the GOAT.
This man is a legend!
I have a question: at the beginning of the video you said that integration by parts is not possible because "we don't have another part" - however; we could have used "1" as the other part - sometimes we do this with integrals like the integral of ln(x). Not that we should use integration by parts, since taking the derivative or integral of that would make the integral more complicated, or that I don't like your videos (I personally LOVE them), I just thought that it would be an important clarification to make :)
I looked up this video because I don't know which side the x should be at. Now I know but I still don't get why we have to pick the x on that specific side?
#ADA #Handicap not only option for 135.7-Eleven
No homo. Such a great and good looking professor. Funny, kind, patience, good looking. What a great professor you are sir. Thank you for the knowledge.
Thanks for your great lecture
great video
You're the man
THANK YOU..
YOU'R THE BEST MAN...
Thank you SO SO MUCH. You saved my poor soul~~!!
How is (2√secθ^2 -1) equal (√4secθ^2 - 4)? At 1:20:44. Online calcs say these are unequal.
Excellent teacher!
14:06 continue tomorrow thank you professor Leonard
brilliant excellent good superb
Dear sir, is it possible to make,for instance, a u-substitution where the differential (du) is located in the denominator? Thank you very much for all the material you posted online.
Unfortunately, no. You need to spot the derivative of the inner function in the numerator, for u-substitution to work.
You factored out the negative, but you forgot to change the sign of the 3 ,
dude u really kick ass, I am taking this class now and why cant you just be my teacher
I never understood why for a²-x² we can’t use x=acos(θ)