Integral by completing the square, and u sub, calculus 2
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- Опубликовано: 3 июл 2024
- Integral by partial fraction & completing the square,
check out @bprpcalculusbasics for more calculus tutorials.
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I always find it beautiful how trig functions can appear seemingly out of nowhere
Literally out of nothing
One person's definition of beauty is another person's definition of horror. You math people are strange, especially when you start talking about enjoying Rudin.
@@Daniel-aaaaa for real
there are hidden inverse trig functions among us
@@SISKCERTWaJaVlogs TRUMP 2024🇺🇸🇺🇸🇺🇸 IMEACH SLEEPY JOE!
You are an excellent teacher. Please dont stop.
That's what she said
I think one of your best quality is to teach math with a smile.
PS: very gratefull for the D.I. method and this square trick
dude that’s a great problem, big thank you. such a joy of a personality too, super enjoyable to watch. i hardly comment on tutorial videos, but this one was truly exceptional. I will remember to subscribe in the fall, but would like to forget that calculus happened for a few months after my exam, so I’m going to keep calculus related stuff out of my sight, haha. Great stuff though.
Thanks!! I am very happy to help and hear this!
What happens at 3:40 helps solve a HUGE problem for me that's been bugging me for hours. Thank you so much!
Your timing is impeccable!! I'm being tested on this in 5 days, thank you Laoshi !!
: )
@@blackpenredpen wait, your name is laoshi?
Sultan El Shirazy It means teacher in Chinese. lol
This has been the most helpful video ever. I couldn't find this method online, and Webassign tells me to use partial fraction decomposition when I can't, in cases like this. Thank you so much!
Thankyou!!! Love your energy while teaching.
This video has helped me immensely
man you're the best it's a long time i check to resolve the exercise like this! And you do it well thx for all
studying for my midterm and this was super helpful! thank you for sharing your knowledge with all of us :D
Tysm. I was lost on trying to integrate with an irreducible squared function on the bottom. I completed the square, squared the result, and followed your steps to completion.
You’re a real one 🙏
I love your videos, MathPapa! Very helpful
Thank you very much for this video, man. You always make it look easy.
You can exploring this learning for me. Thank you so much
You can also multiply the top with 2 and multiply the whole integral with 1/2 , then substract 6 and add 6. Then you get 1/2(int((2x+2)/(x^2+2x+5) + 6int(1/(x^2+2x+5)))
You are very good at being a tutor.
Wow finally I got someone who did this problem...... U helped me a lot
Man ur such an amazing teacher I mean I couldn’t understand this no matter what and it took u 10 min to get it into my head . Love the way you write though 😂
actually the best math video ive ever seen
Im a Calc 1 student who has just barely begun differentiation, but you sir, have Madeira possible to follow along to a more complex integration topic 👏🏻
I don't usually comment on RUclips, but you are a lifesaver!!!!
FANTASTIC !!! Thanks a lot
keep going. excellent work!
you are such a sweet guy, you make me smile while learning calculus. thank you so much, brilliant video
Great job ! very helpful.
This was a great video, thank you.
You are a magician!
excellent explanation.
Piece of cake! I remember learning this in elementary school. Of course all of us kids did it so much faster, but we didn't have to explain it on the blackboard. Great review!
Bruh
Elementary school,lol😅🤣..
way more helpful than what we learn in class
Mans is so happy. Thanks✅
Beautiful..brother...love your videos
Beautiful sir ..... Thanks
This guy is amazing
You are a very good teacher but you upload very frequently and my brain can't handle so many integrations.
Its not you, its me.
Wow! You're awesome!!
Greate Video...thank you very much
9:39 "Finally, finally" lol😂
Why is it that if I do polynomial long division before completing the square I end up with a different answer of (1/2)x^2 - 2x + (13/2)arctan((x+1)/2) + C ??
Alternate method:split numerator into differntiation of denominator+constant
In this way you get 2 solvable integrals(1st one is too easy, 2nd one-just integral(dx/quadratic) which u know how to solve....)
Put x+4 = A(2x+2)+B. Find A and B.
Then split in to 2 integrals, that is easy method
Lol yeah that's how I did it. Learned it from integral calculators. You always gotta be looking for a way to make the derivative of the denominator in the numerator.
Question:
what is the integral of-
(x+4)/(x^2+2x+5) dx
*looks at title*
yes u should sub to this channel
: )
I posted this comment a year ago when I haven’t taken Calculus and didn’t understand the title. Now that I completed my first semester of Calc AB and looking back, it’s just so silly I made this joke from lack of knowledge 😂
@@HeyKevinYT lol
Thank you man
Thanks Buddy!!!!!!!
The complete the square to U sub to trig sub pipeline is real 💀
I enjoyed that 👍
That is an easy integral. We used to solve hundreds of integrals of fractions of higher degrees. You only have to write the numerator as a derivative of of denominator maybe multiplied by some constant, and then write the rest as a fraction of the form : λ/[(x - x(0))/a)]²+1. (In our case Δ
excellent I really understand how to integrate square numbers from Papua New Guinea
I am from India but able to understand his explanation is very easy Thank you so much 🙏🙏🙏🙋♀️.
Sir🙏
12th?
Could you use usub and take the whole denominator and take the x of the numerator?
He speaks amazingly , looks like different dialect
Thanks sir, but for me. I always use u=x+1
x=u-1
Substitute to x+4
= u-1+4
=u+3 which is just the same
My professor doesn't want us to remember the integrals of inverse functions, rather we are supposed to do a trigonometric substitution every time.
the only question i have why we take 5:37 w=u^2 +4 and not just u^2
(g.i.f of x+y + gif of x-y )= 5 x ,y are greater than or equal to 0 x>=y find enclosed area .
bro plz solve this plz plz plzzzzz 😐😐
please link the video you referenced of you explaining the inverse tan formula. cannot find it and it doesn't make sense
You can differentiate it to show that it works in reverse.
integral 1/(x^2 + a^2) dx = 1/a*arctan(x/a) + C
Let C = 0, and let y = 1/a*arctan(x/a). We're interested in showing that dy/dx, equals the original integrand.
y = 1/a*arctan(x/a)
Multiply thru by a, and take tangent of both sides to clear the inverse tangent:
a*y = arctan(x/a)
tan(a*y) = x/a
Use implicit differentiation to find dy/dx:
d/dy tan(a*y) = a*sec^2(a*y)
d/dx tan(a*y) = a*sec^2(a*y) * dy/dx
d/dx x/a = 1/a
Thus:
a*sec^2 (a*y) * dy/dx = 1/a
Solve for dy/dx:
dy/dx = 1/a^2 * cos^2 (a*x)
Recall the original value of y:
dy/dx = 1/a^2 * cos^2 (arctan(x/a)) = 1/a^2 * a^2/(x^2 + a^2)
Thus:
dy/dx = 1/(x^2 + a^2)
Which is the original integrand, we were originally trying to show, would integrate to 1/a*arctan(x/a).
Let x²+2x+5=z²
Then (2x+2)dx=2zdz
Then (x+1)dx=zdz
At 6.54, isn't this just in the form of differential of the denominator over denominator, and thus can be integrated directly? Great video as usual, thanks.
Which one are you talking about, mate?
i loved the way he solved the problem with smiling face...while me trying to solve that with stress face😔
Great video. My only issue is that the first part of the integral split can be done using reverse chain rule, no usub needed (it's messy and long). Plus the second half formula can be derived from reverse chain rule.
Can somebody tell me how that substitution happened? How that tan-¹ came up, and if that is arctan or csc, or tell me the video where he talks about that substitution in more details
It is arctan.
d tan x/dx = 1+tan²x, so d arctan u/du = 1/(1+u²) and d arctan(u/a)/du = (1/a)/(1+u²/a²) = a / (a² +u²)
How Nice !
7:01 which video of yours should i watch to know how the formula work
I responded to Kieran with a proof for how this formula works.
It always worries me that you use the 'inverse' notation to write a 'reciprocal' function. How do you then write the inverse of the latter? Why not use the beautiful notation arctan or arctg (or even atan or atg)? Idem for arcsin and arccos.
For the other math-challenged, Can you give some everyday examples of how math like this is applies in and used in real life?
Depends on what you call 'everyday' and 'real life', isn't it?
For example, can you tell me how watching football everyday is useful in real life? Or music? Or art and poetry? Or history? Or quantum physics?
As to the original question:
when you have a bunch of data and want to adjust it to some probability distribution, it is useful to compute d ln f(x)/ dx and if it has the form (ax+b)/(xx+cx+d) [Pearson distribution] then you can use the four first moments to determine a f(x) with the same moments.
I use that "every day" in "real life".
It also appears commonly in mechanics and in optimisation problems.
Does anyone know which one of his videos is the trig video he mentioned?
I responded to Kieran with a proof for how this formula works.
0:17 we have a Russianal function here
Every high school kid: math again🤮
Every engineering student: 3 math classes per week😝😝😝
Somehow you learn to love it, it becomes an addiction🙈
Hi. What if the denominator has a 2 in the x^2, i mean what if the denominator is 2x^2+2x+5 same function in your video.
You can always pull a constant out in front of the integral.
If I were integrating 1/(2*x^2 + 2*x + 5) dx, that's how I would solve it.
Given:
integral 1/(2*x^2 + 2*x + 5) dx
Pull out a factor of 1/2:
1/2*integral 1/(x^2 + x + 5/2) dx
Complete the square:
x^2 + x + 5/2 =
x^2 + x + 1/4 - 1/4 + 5/2
x^2 + x + 1/4 = (x + 1/2)^2
x^2 + x + 5/2 = (x + 1/2)^2 + 9/4
Reconstruct:
1/2*integral 1/((x + 1/2)^2 + 9/4) dx
Multiply by 1 in a fancy way, to turn the 9/4 into 1:
1/2*integral (4/9)/((x + 1/2)^2*(4/9) + 1) dx
2/9*integral 1/((2/3*x + 1/3)^2 + 1) dx
Let u = 2/3*x + 1/3
Thus: du = 2/3*dx, and thus dx = 3/2*du.
Replace dx with 3/2*du, and complete the substitution:
2/9*3/2*integral 1/(u^2 + 1) du
The integral of 1/(u^2 + 1) is arctan(u). Thus:
1/3*arctan(u)
Recall u, add +C, and we're done:
1/3*arctan(2/3*x + 1/3) + C
i dont get why u do a "w" substitution...why can you just apply the formula and plugin u?
This is a quick example to work out: ruclips.net/video/HYEGfrbMZm8/видео.html
Wooooo I got it
Integrate cos (x^2) . Its seems to be easy but its not. Please solve it sir.
not elementary function
The antiderivative of that function cannot be expressed in terms of elementary functions. However, you can use the Maclaurin Series for cos(x), and just replace x with x^2.
You can solve this using trigonometric substitution too ❤️💕😍🥰💯🧸
tried it it dosent work out the same and its much longer way to solve the problem...
Yes, it works out (for the second integral), but it is a bit longer.
substitute x+1=2tanθ then integral gets so much easier than this
Where are you from brother? I wanna join your world of mathematics
He's from Taiwan, and lives in California.
I keep hearing (th th ... )Th throughout the video
I wish I was good at maths to understand this, it's all foreign to me....
dude left out the constant term until the last line... why?!
Because he can. All that matters with the integration constant, is that your final answer has it, to represent all possible integrals. If he included it at every step along the way, he'd have to reassign C as he combines multiple constants. It's easier just to let C=0 at intermediate steps, and wait until the end to assign a master +C.
This is only valid for single integrals when you do this. When you do double and repeated integrals, you have to account for +C at each of the original given integral layers, and you end up with multiple constants of integration, which may both have significance for the application of integration. Such as +C1*x + C2.
YaY!!
nice vedio••••••🥰🥰🥰🥰🍊💕
I LOVE U
Me:
Enters video
Sees 3 colors
Unsubscribes
Lol just kidding :)
this is a cruel subject
Goodness, I'm the 3rd commenter!! #YAY
Oh man I was just introduced to the "W world" and i don't know if I wanna meet this b**** wahahaha
that's a really good example, beautifully done!