integration by partial fractions with an irreducible quadratic factor
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- Опубліковано 21 жов 2024
- Learn how to do this integral of a rational function by using partial fraction decomposition. This integration has a cubic denominator and we will see how to factor it to get a linear factor and an irreducible quadratic factor.
For more calculus tutorials, check out my new channel @bprpcalculusbasics
/ justcalculus
I got a perfect score on my first Calculus II exam thanks to you. I think I owe you my tuition now.
Awww that's so awesome to hear! No you don't owe me anything.
"Thanks for watching" (with Dr. P's voice"
**happy integral noises**
If everyone who's studied calculus from bprp gives him even 10$, then bprp would be a millionaire.
Rather than traditional integration techniques, Partial Fraction Factorization was the main principle regarding this very integral..
Thanks for presenting to us!!
😊😊😊
bprp, I started watching your videos back in 2016 when I was first taking Calculus 2. Now in 2019, I am an LA for a Calc 2 class myself and I use your videos for inspiration to help my students study and give them a little challenge here and there ;D I’ll definitely ask them this question tomorrow since they have just finished going over partial fraction decomp. Thank you so much for the past few years of tricky math content, and for future content to come :D
Christian Rodriguez thank you! This is so awesome to hear!!
Hey, they never taught me that way to factor things out you did at the start of the video! Thanks!
This is the only channel on YT where I can always find the most interesting and complicated things related to Calculus. Most of the videos are useless since extremely easy and every student preparing Calc 1 or 2 already knows all that stuff.
Thanks for this upload! Partial fractions can be confusing, and you made it simple.
Hope all is well.
Thank you dear teacher, b🖋️Pr🖍️P ❤️
best math class in ages
Instead of doing the long process of partial fractions, you could have notices the +8, which is a multiple of +4, and the -3x, which is a multiple of -3. Then ou can quickly check to see if it works out with the x^2 term, which it does here, and you can seperate the fractions
Exactly my point
Two random points are selected inside a cube of 1 unit. What is the average distance of the two points?pls solve this
There is a mind your decisions video on it
Pressure cooker is garbage tho
@@Ethan-mj6wy lol
@@VaradMahashabde I thought this video is about the average distance of 2 random points in s square.. I guess to calculate this in a cube ist even harder
Still the same kind of answer ish
But the low quality of french smallwalker is unmatched
Thank you so much!! You are the best, love the vibe!!
Perfect explain 👍👍
I love calc 2! I like your videos
Thanks!!!
Good job.. how the value of A=2 ?why x=3?
Thanx.
I am an electrical engineering student, thank you very much for sharing this method! MABUHAY!
from Philippines.
NICE
I was trying to factorise the numerator😅
Do you have a dedicated partial fractions video?
Not a bad idea to check if 3 is a solution to the numerator. Though you should see that one won't work.
I also want it.
Δ = 3² - 4(3)(8) = -87 < 0
∴ 3x² - 3x + 8 is not factorable.
@@gordonchan4801 You also could use modular arithmetic mod 3 to show that it has no integer solution.
@@sugarfrosted2005 and in fact, no real solution.
Very valuable technique for advanced level math, i.e. Differential Equations.
R u preparing for JEE advanced
You could rearrange numerator also
Thanks so much for this video, I needed it!!!
Thank you for the video.
We can substitute x=0 for C and x=1 for B.
just wondering, if you use the cover up method on x^2+4 with x=2i will it work?
@@JohnSmith-ry5yc But you can change form of complex numbers. So you can get arctan(x) that way.
In fact I've done said integral of (x^2+1)^-1 using complex numbers, and converted the x+i and x-i parts to exponential form.
Great job, mate!! Your presentation is very cool.:):)
We can guess how to express integrand as a sum
3x^2-3x+8=2(3x^2-6x+4)-3x(x-3)
and then in first integral we can sustitute u=x^3-3x^2+4x-12
in the second integral numerator cancel with denominator and we can substitute v = x^2+4
Solved it on my own.
Nice 👌
So brother, I am from Bangladesh. your videos regarding mathematics are interesting. It will be great if you answer my questions.
1. What is the formula of median and why so!?
2. What is the formula of mode and why so!?
It could be in mathematics and also statistics.
Thanks, u rescue my assignment 🤣
-1 < 2*logx(8) < 3, solve for x
Very cool!
Just the algebra alone is so interesting to me
yay, i'm studying partial fractions right now at school
I can't thank you enough.
Would it be correct to just plug in x=0 to the equation to find C? And then find B by plugging any value, like x=1?
After finding A, to get B you can multiply by X and compute the limit to +∞, you easily get 3 = A + B --> B = 1. To get C just X = 0.
Sorry if this has already been said in the comments.
Thanks.....😊
Could you explain why did you replace x with three to find A?
(x-3) is a factor of the second term and therefore substituting x=3 makes that vanish, you could substitute any other real number but it would leave A depending on B, and you would have to substitute the value of B (1) to get A
what if the numerator is bigger than the denominator
is it possible to get an impossible system of equations(i.e. with conflicting values for a single variable) or does it always work out no matter what?
it always works whenever degree of numerator < degree of denominator. Online proofs are available.
Can you do the cover up method by using some complex number for example, and then taking the real and imaginary part on the sides?
Thank you
Can I write the answer as ln(sqrt(x²+4)(x-3)²) + c?
Could we do some partial complex function? I mean for the part (x² + 4) can be (x - 2i)(x + 2i)??
yes
thanks man, shit saved my last life at uni haha
I think It would have been easier to replace X by 0 to find C and then, X by 1 to find B . I did it that way previously showed in your videos and it was faster. Anyway great job as always !
Make videos on binomial theorem plz
Actually, it easier to find C by plug in x=0 since we know the value of A. Then plug in x=1 to find B after we know the value of A dan C
What is integral of √(1 - x³) from 0 to e ?
How can we doing tan(cosX+isinX) when x is 2 or -2?
How about if the denominator is not factorable? Let’s say the denominator is x^3 - 3 x^2 + 4x - 11
Can we have reduction formula of I(n) equals integral from 0 to pi of tan^n (x)dx ?
Love ur vids
Can u publish more semi-hard integral videos
Thanks a lot it really helps, keep it up !
Love u
I have a calc II test on Thursday and just seeing that integrand stresses me out
Integral of Ln|cosx|÷x
Another successful integration 👌
Why don't you do complex cover-up to solve for B and C? It's just good-old algebra as you said ... Algebra 2.
I summon the God of mathematics to heart me
can you try describe to us -> integral (1 / lnx) dx
i am trying to calculate this on calculator and it says " li(x) +C "
What is the mean of "li(x)"
li(x) is a non-elementary function, that stands for "logarithmic integral". By definition, it is the integral of 1/ln(x), which has no elementary integral. It is also called the polylog function.
There are non-elementary integrals, associated with each of the original transcendental functions, when they are put in situations where they have no elementary integral.
Sine integral: Si(x) = integral sin(x)/x dx
Cosine integral: Ci(x) = integral cos(x)/x dx
Exponential integral: Ei(x) = integral 1/x *e^x dx
Logarithmic integral: Li(x) = integral 1/ln(x) dx
The answer to this integral is
Ans:- ln(x^3-3x^2+4x-12)-ln(x+4)/2 + ln(x+4) + C
what konwledge do you use
What abount an irreducible QUARTIC?
thank you so muuucchhhh
hey blackpenredpen pls tell how to integrate
1/(x^4 + x^2 + 1)^1/2
int(1/(x^4+x^2+1)^0.5)dx
=int(1/√{(x^2+1)^2})dx
=int(1/(1+x^2))dx
=tan^-1(x)+C
@@alex_schwartz it does tho, (x2+x+1)(x2-x+1)
@@NeelTigers Correct, done by long division where x^2-ax+1 is the division factor i guess?
Neat trick
how you find A
Can you do the x/x^2+4 in differentiation by substitution step by step for me pls I didn't get that thanks!
Let u = x² + 4
du = 2x dx
dx = du/2x
Integral of (x/x²+4) dx = integral of (x/u).(du/2x)
See that x/2x cancel to be 1/2, therefore
Integral of (1/2).(1/u)du which gives 1/2 ln |u| = 1/2 ln (x²+4)
Note: Since x²+4 cannot be negative, you can use parentheses instead of absolute value.
That's it
@@juliocesar4442 thanks cheers
Can someone teach me how to do or no solution
x=(y^x-1)/y-1 in term of x
Where do you live sir?
: )
Somewhere in middle of taiwan
Hrushikesh Naik Somewhere on earth!
Ive seen the meme where you slap a chicken 1665 m/s to cook it. Can you show us how to calculate that speed
I assume that the conversion rate from kinetic to heat energy is 100%
So Heat required to cook chicken is x J
m is mass of your hand
mv^2/2= x
v=(2x/m)^1/2
I guess is this is how it works
Please can you sopve this question
(2x)exp(ln(2)) = (3y)exp(ln(3))
3exp(ln(x)) = 2exp(ln(y))
find x and y
2x*exp(ln(2)) = 3y*exp(ln(3))
2x*2 = 3y*3
4x = 9y
y = (4/9)x
3*exp(ln(x)) = 2*exp(ln(y))
3x = 2y
y = (2/3)x
@@EnzoSantos we require answer for x and y in numbers as in 1,2,3.1456 ... Etc .
@@EnzoSantos in formula number 2 is y=1.5x ... the only one answer is 0?
@@saharhaimyaccov4977 i don't get the question (and i know that you aren't the person who asked it), is this a system of two equations and what does mean exp(x), is exp(x)=e^x for all real x? if so, then you said that answer is 0 meaning that solutions for first and second equations intersects only at x=0 (and y=0), shouldn't it be then (0;0) (or maybe (0,0)) btw in second equation x>0 and y>0 as ln(x) and ln(y) exists (i mean we are in real numbers, aren't we?) so if this is a system then maybe answer should be Ø, am i wrong?
3:15 is it just me or is that the Doraemon theme song?
I did this by getting a robotic blue cat from the future to do all the work for me
How to integrate sqrt(x^2+16)
Trig sub
Let x=4tan(u) and dx=4sec^2(u)du so you get
integralof[sqrt(16tan^2(u)+16)*4sec^2(u)]du
Then factor out the 16 under the square root so you end up with
4*4*integralof[sqrt(tan^2(u)+1)*sec^2(u)]du
and here you recognise a trig identity : tan^2(u)+1=sec^2(u) so finally, you just have :
16*integralof[sec^3(u)]du
If you have issues to solve that last one, bprp made a video about it.
At the end, just make sure you go back to the x world and you're done
@@メ乇しム尺
I did not recognize the identity. Thanks for help
@@lordofcastamere9376 You're welcome (:
7:46 ...parenthesis!!
How you solved for A doesn't make sense. What do you mean you cover up x-3 and then sub in 3. That doesn't make sense.
That's the way the coverup method works. You cover up the corresponding factor to the coefficient you're finding, and then make what you covered up equal to zero.
Given a general rational function of:
F(x) = N(x)/[(x - p)*R(x)]
Where
N(x) is your numerator polynomial
p is the pole in question
R(x) is the remaining polynomial of your denominator, which doesn't equal zero at x=p.
When you set up the partial fraction expansion, you get:
N(x)/[(x - p)*R(x)] = A/(x - p) + Q(x)/R(x)
A is the coefficient we're trying to find, and Q(x) is an arbitrary polynomial on top of the fraction for the remaining terms.
Multiply through by (x - p):
N(x)/R(x) = A + Q(x)*(x - p)/R(x)
In the limit as x approaches p, the factor (x - p) approaches zero. This annihilates the term that started as Q(x)/R(x). We are left with:
A = N(x)/R(x)
Now we can plug in x = p directly.
A = N(p)/R(p)
And this is what Heaviside's coverup method prescribes. It tells us to cover up the pole associated with constant A, and evaluate what remains at the corresponding pole. Since R(p) doesn't equal zero, we can evaluate this directly.
You are uploading videos after long intervals. Pls upload faster. I am your regular watcher.
Isn't it better to focus on videos instead of rushing it out?
Lmao dude this guy is one of the most active youtubers i know chill a bit
@@gregorsamsa9762 He used to make videos faster than the present rate of making videos. So i requested him to make videos faster as he did before
are you serious? he is churning out videos at an incredible speed, and he still has a day job. do you want him to burn out?
You might be able to kick calculus in the head but algebra is gona give you a right hook to the jaw. Algebra>calc.
Seriously !
I don't know what the 3 dislikes for !
Cool
isenit
Noice