Integration Using method of Partial Fractions

[@Skystrike70]

0:34 'I will fix this soon" -JMT 2008

[@jindassasin]

it's people like you that give me hope for humanity

[@ilikechess1]

To any future viewers: it's Dx + E because 1 + x^2 is irreducible.

[@merensow200]

!

[@OkieDoke405]

Lot of profs have a way of complicating relatively simple concepts - don't know where I'd be without you. Thanks for taking the time to do all these. Saint patrickJMT lol

[@patrickjmt]

+Malcolm Lewis ;)

[@Neverdii]

thanks for being one of the only people willing to volunteer their time to help others with math! I wish there were more people motivated by helping others instead of by money like you...it's a huge help man, thanks a ton

[@blackcard00]

Thank you so much. You talk to me rather than repeat from the book and show answers, unlike my professor. It is much easier to pay attention and stay awake to. Thanks a ton.

[@patrickjmt]

@MillerTivo x is a factor of x^2, so there is an 'x' factor

[@patrickjmt]

yes, absolutely. a few others pointed this out; i have added annotations pointing out the mistake. thanks for pointing it out!!

[@patrickjmt]

@DashofDosh you can use either method

[@patrickjmt]

@pricesalem no problemo

[@patrickjmt]

that is the partial fraction decomposition for linear factors... i have a video on my website explaining that step if you feel the need to check it out!

[@bryandPR]

no patrick no! theres a few of us who are really thankful for these videos!

[@patrickjmt]

@janedoherty101 do you mean a square root instead of a squared expression? if so, that totally changes the problem. it just depends on the problem at that point. in the case you mentioned though : (x+1)^1/2, you can do a rationalizing substitution in some situations

[@patrickjmt]

@Kratos0909 glad you like it ; )

[@Ichneumonidae2]

@AlecWangable When decomposing anything raised to the second power you have to repeat the variable. So, if you were decomposing (x+1)^2 it would be A/(x+1) + B/(x+1)^2. The reason you do this is because if you only write one fraction (e.g. A/x^2 w/o also adding constant raised over a denominator "x") you will be missing a factor of whatever is being raised. Try decomposing 1/x^2(x+1) by only writing A/x^2 + B/(x+1). You'll come up with the wrong answer b/c you would be missing a factor of x.

[@k0tiak]

Are you sure the last term isn't Dx + E in the numerator? Usually if (b^2 - 4ac) for the factor is negative, you use Dx + E (or whatever letters/variables your using) rather than just a D. (0^2 - 4(1)(1)) = -4 which is < 0. For (x^2+1)

[@alexprocel91]

Anyone know why he used D instead of Dx+E at 0:32? I thought you only use a constant variable when it's a linear factor.

[@brunocalander1907]

yeah I tought it was Dx+E too.

[@bravosix8171]

The factor is linear take x² as (x+0)². So the easiest way to do is to treat it as repeating linear factor

[@GuaranteedPurity]

there are only 3 terms in the denominator but you added extra one why is it

[@sirsideways]

@MillerTivo it is considered a repeated factor, giving A/(x) and B/(x)^2 as with all repeated factors, however in this case the factor (x) is x itself instead of something like ax + b. It is not a linear term, or an irreducible quadratic hence why that expansion would be incorrect.

[@djtsuna19]

i started watching your videos last semster its differential calculus, and now im on integral calculus. hopefully you can help me for the my last two math subjects. differential equation and advance math. thanks patrickjmt! you're a real help! im taking bs-ece btw. :D

[@gotahope]

Thanks mate, this is such a good help 3 days before my HSC exam :)

[@patrickjmt]

@wintermonie ok

[@melisamani4299]

when X^2 is in the Deno don't we need BX+C at the Numerator?

[@the_eternal_student]

I had to look at Nancy Pi's channel to see that this involved a tangent substition and why. I came to this channel based on her featuring your channel, but although you provide more content, so far I think you can learn some things from her about explaining everything.

[@anyoutubehandle]

nah

[@nataliatkalangula904]

Thanks Patrick. your video are of much help for my studies.

[@suplex57]

could you further explain the arctan rules and where can i find them?

[@buddydog1956]

I'm studying for the FE Civil.... thanks for your videos! my favorite, nonetheless was ODE's! ... PDE's (don't ask!!! they're killers!) But Keep the work coming, You're the Man!!!

[@ahsanshah7626]

This video is so helpful!

[@sierraalbin4669]

This has helped me so much! Thank you for sharing your knowledge!!! Is there a link to look at the answer with the Dx + E

[@mrapollo13]

For anyone wondering what the integral of the final dx+e term is, if my math is correct it should be: 1/4ln(1+x^2)-1/2arctanx. treat the brackets as absolute value signs.

[@festexi24]

Thank god for this video, it looks so simplistic but the lack of variable up top just sent my head spinning!

[@jonpbody]

i've learned so much from you man, thanks so much for making these videos for us.

[@axelasdf]

... I did it all without the "trick" at the start where you set x=0 and x=-1, where I just multiplied it all out and solved using equating of the coefficients, and after all was done, I got "1/x^2+-1/x". Did I carry this out correctly? Would you have gotten the same if you had multiplied out completely / didn't substitute in your values that you got from your "trick"?

[@patrickjmt]

@ryezizzle none. i studied.

[@the_eternal_student]

Why did you use arctan for 1/1+x^2? Is this some trig function you have memorized or was it deduced?

[@aimanal-akwaa110]

You are the best, do you have the link for the corrected Dx+E Thanks

[@TayIsCray]

is the updated video on the account? I do not understand the Dx+E.. SOS(:

[@Savvy07]

Correct Ans : (Dx+E) *[1/4 ln(1+x²) + 1/2ln(x+1) - ln(x) - 1/2tan⁻¹x -1/x ]*

[@jemcel0397]

Where did you get A/x? Confused

[@Jubjubben]

+Jem Celespara oh wait I think I see your question 1/x^2 => a/x + b/x^2 that's a rule called "repeated linear factors", therefore if you had 1/x^4 you would get => a/x + b/x^2 + c/x^3 + d/x^4

[@jemcel0397]

+Ben Cruse oh. I get it now :)

[@smitpatel4022]

Hey it should be Dx+E instead of D at 0.36 sec. Please confirm same.

[@bravosix8171]

Nope it is not a quadratic eq. Take it as a linear equation (x+0)².

[@Alive4Metal]

where does the single x term come from? im confused

[@DashofDosh]

Okay. sooooo I used Dx+E in the beginning, so how would you go about solving for A, D and E later on in the problem? Do you use elimination or substitution?

[@jasvinsingh9383]

Why isn't it Dx+E instead of just D? It's the denominator one polynomial higher?

[@WorldOfDeepThought]

I solved it with Dx + E and got: A = -1 B = 1 C = 1/2 D = 1/2 E = -1/2 Can anyone confirm this? My solution for the integral is then: -ln(|x|) - 1/x + (1/2)*ln(|x+1|) + (1/4)*ln(x^2+1) - (1/2)*arctan(x) + c

[@Dalton1294]

No, 1+x^2 cannot be reduced by the method of reverse foil, the roots are imaginary Proof: x^2+1=0 x^2=-1 x=sq rt of -1 x=+or- i

[@kathleeneusebio44]

does the roots is applicable even your numerator is a polynomial as well?

[@dante4838]

What if your given limits in the integral? Like the integral from 1 to 5 sqrt of 3? Then the fraction.

[@danagamer]

Since the original rational expression has three products in the denominator, could you have split it into three fractions, then integrated them individually ?

[@The3jji]

@lilyvoon10 @lilyvoon10 Well, the denominator has a (1+x^2). So, the nominator should be (D x+E) Refer to previous videos if you don't understand

[@axelasdf]

Where Can I find the corrected video?

[@anwarabdulkadir4461]

@patrickjmt at 7:41 when i took A+B =0 i got A = -1, why?

[@rolito2243]

is wrong. the factor must be (DX+E)/(1+x^2)

[@patrickjmt]

ha. i often find myself disliking humanity so much, i wonder why i make these, and wonder if one day i will just take them all down... : )

[@KioraNaidoo]

Seriously thanks a million!!! Actually understood!

[@King-ue4dg]

Thanks dude you,re video is really helpful.

[@lilangel0072]

@chadbowman0 I took calc 2 last year...and it was easy, this also happened to be the easiest part in the whole course. Just because u found it a challenge doesn't mean it is one.

[@MultiMoe91]

is the corrected video available? if yes please link me, appreciate it :)

[@marforiyaa]

why didnt you make D, Dx+E?

[@matmadara3]

not Dx+E??

[@beckwilde]

+Mat Madara why is it Dx+E can someone explain

[@jorinnahuang475]

+Mat Madara yea..that's what i'm thinking... anyone knows why?

[@matmadara3]

***** i forgot why it should Dx+E XD

[@diaagamel7946]

+Rebecca Chen bcuz the X is to the square power , when its 1+x^2 we should use Ax+B for example but if it is (1+x)^2 (the whole bracet squared then it goes like A then B and so on ..

[@beckwilde]

+diaa gamel thank you everyone for the explanation

[@siana2nv]

heyy, just a quick question!! how do u make/record these videos?? like whats the setup?? plz respond, thanks!

[@JONATHANGOPIE1]

keep your good work up man

[@janedoherty101]

what if instead of the denominator having a squared expression it had a squared one? ie (x+1)^2 vs. (x+1)^1/2?

[@shiftanimations1557]

Sir patrick you are the star!!! But there's something bothering me about my professors teachings... usually if the denominator is a quadratic equation then the numerator is for example Ax+B right?cuz that's how you, the book, and my stock knowledge does it... but why did he said it should be : if the denominator is x^2+1 take the derivative of it which is 2x then use that instead which he means it'll become A(2x) + B...is that wrong? pretty sure it is but please let me know your thoughts about this or let me know as well if i had not explained the situation clearly

[@JessicaViduya]

Since Dx+E isn't included, I tried to find them myself. I got A= -1 B =1 C =1/2 D= -1/2 E= 1/2. Is that right?

[@patrickjmt]

i'm too lazy to work it but maybe someone else can verify!

[@rutwolvaardt792]

I did the calc and got the following A= -3/2 B = 1 C = 1/2 D = 1 E = -1/2

[@pricesalem]

good example, very helpful, clear and concise explanation. THANK YOU!!!!

[@lzwanlsmail]

why it is that '1+x^2' is equals to tan x? i forgot the rule.. can someone explain why?

[@saswatmishra6917]

It doesn't equal to tan x,it equals to the arctan x. If you differentiate arctan x, you get 1/(1+x^2). You just have to memorize what happens if you differentiate arcsin, arccos, etc. I don't know the proof of how you get from one to the other to be honest

[@Reinout95]

I owe my Geology degree to you. :)

[@patrickjmt]

ha, but none of my videos are about rocks!

[@EconAtheist]

patrickJMT BRO, MATH ROCKS #sorrynotsorry

[@tlotlegomokalake6347]

😅 I'm also doing Geology.

[@hyky68]

ehh so.... in the beginning, why do we need to have the a A/x term when there's really no single x value in the original equation?

[@mohibfayyaz4582]

somebody helps me. Whats going on after 6:35 ????

[@patrickjmt]

you are doing what is known as 'equating coefficients'. on the left side, there is no x^4 term which you could write as 0x^4. You take the coefficients of the x^4 term on the right side, which are A + C and set them equal to the coefficient on the left side and so you get A + C = 0. you do the same with the rest of the terms. rewatch it as i explain this :)

[@mohibfayyaz4582]

@@patrickjmt i got it .. Thanks

[@mohibfayyaz4582]

@@patrickjmt i might be wrong but... in previous videos you told that... when there is quadratic function in denominator, Take (Ax+B) in numerator. But in this video you take simply (D) over (x^2 +1) Please explain if i am wrong.

[@iPooAlotz]

wtf just happened

[@silversteel9161]

Layzi HAHAHAHAHA !

[@SehamArad]

i think it was a genuine question by @monie otanuho...dont think they realised there is so many other Patrick videos explaining how to do partial fractions...Patricks response was hilarious though..these vids are great!!

[@taymurahmed7119]

+patrickJMT I just love maths.... cause u made me to think that way Patric Sir !!! :)

[@patrickjmt]

ha! do not worry : ) 99.99% of the youtube comments have been so amazingly nice that my faith in humanity is being restored : )

[@anzatzi]

This is a good problem which requires solving amicable x vals and solving with coefficients. The amount of algebra is nearly overwhelming (for me)

[@KalCrazyFJ]

what if we had dx as the numerator for eg. dx dx x^3+x help please?

[@lucile1010]

whou you're so cool! that's gonna help me alot when im being lecture tomor. ='DD PS;/ I love your writing!x

[@lilangel0072]

@chadbowman0 there's waaayyyy harder stuff in Calc 2 involved, this stuff isn't only easy by comparison but it IS the easiest this class is gonna get...honestly take it from me, if u can't do this, don't bother with the rest of the course.

[@lilyvoon10]

why would it be Dx+E?

[@puteracamaro]

I agree with what you are saying.

[@adityachaudari4321]

2x/(3x^2-2x+1) esko kyase expand kare ?

[@trevort888]

why isnt it just D?

[@martymcfly347]

He isn't lefthanded. He learned to write in mirror images and then he just flipped the video. You can do it too!

[@mattarcarese6136]

Correct Answer: -( ln| x | + ( 1 / x ) ) + 1 / 2 ( ln| x + 1 | ) + 1 / 2 ( [ ln| x^2 + 1 | ] / 2 - arctan(x) ) + C

[@patrickjmt]

yea, partial fractions is a long procedure unfortunately! hope it makes sense though!

[@lilangel0072]

@wintermonie r u serious? he was to the point and explained it nicely, plus this is the easiest stuff u're gonna learn in calc, if u don't get this then maybe u should reconsider ur classes.

[@nealogz]

OMG thanks a lot.

[@emilrusev4552]

definite integral b=1,a=0 x^2sqrt(1-x^2)

[@patrickjmt]

ops yes,i will have to fix this!

[@Me63422]

shot buddy. You should apply to be a lecturer at some top notch university. Maybe university of pretoria. hint hint. Would make my life a lot easier.

[@patrickjmt]

almost impossible to get a job without a phd. if you manage to find one, the pay typically sucks and you only get a one year contract if you are lucky. after that, you may be back on the street again. of course, not many jobs have security, but at most of them, when you get hired, you can expect to be there a while unless you are a total screwup or there are massive layoffs.

[@Me63422]

oh well. this seems to suit you fine as well.

[@Hco1732]

patrickJMT you should tutor on insta edu !! ...please haha

[@No_Englaisy]

patrickJMT Oh my God that is depressing. Then why am I studying? What are you doing for a living if you don't mind me asking.

[@wynandvanwyk9861]

The Ginger Man Tuks of Niks.. ;)

[@dizsun7428]

shouldn't a/x be ax+b/x since the x is squared. then the next term be cx+d/x^2

[@ryezizzle]

what god did you pray to to get so good at math?

[@lilangel0072]

@chadbowman0 then why r u searching this video? i see they didn't teach u well back then.

[@jacqulynchristian4299]

I dont get Partial Numbers At All :(

[@noepineda1673]

Great

[@metalfreak1892]

heyy mann... im a guy so i cant marry you..but i'd hook u up wid one of my female friends.... that helpedd soooo muchhhh THANKS

[@fttank]

nice stick people drawings :D

[@chadbowman0]

@lilangel0072 That's total BS, this stuff is a bit complicated. Because of the algebra involved.

[@marwaniii]

patrickJMT > Khan academy

[@Wind_slash0]

oh how i loathe partial fractions... makes integration easier, but finding A, B, and Cx+D is a pain >_

[@skyfaze]

you can donate to him on paypal!

[@dianaserac6727]

got it :))) !!!!

[@ahmedsuhyl]

most of the time we plug in 0, -1, 1