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Abhisar
 one year ago
Integration Help!
Please guide through the steps
\(\huge \sf \int\limits_{0}^{4} (6t8)(38t+3t^2).dt\)
Abhisar
 one year ago
Integration Help! Please guide through the steps \(\huge \sf \int\limits_{0}^{4} (6t8)(38t+3t^2).dt\)

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imqwerty
 one year ago
Best ResponseYou've already chosen the best response.0ok 1st we need to open the brackets :)

imqwerty
 one year ago
Best ResponseYou've already chosen the best response.0wait....the method with which i did it was really long but i got the answer 176 wait lemme think of a shorter one :)

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0hmmm. I can do this... but as @imqwerty is doing. I don't need to interfere. If he has trouble then I will jump in @Abhisar

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1You mean like this \(\sf ∫u v dx = u∫v dx −∫u' (∫v dx) dx\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4Well, we know that if you have something like an abstract case below: \(\large\color{black}{ \displaystyle \int_{a}^{b} \color{red}{f'(x)}~{\rm G}\left(\color{red}{f(x)}\right)dx }\) f'(x) is just some other function that is derivative in relation to the f(x) part, then you set \(u=f(x)\) \(du=f'(x)~du\) \(x=a~~~\rightarrow~~u=f(a)\) \(x=ba~~~\rightarrow~~u=f(b)\) and you would then have: \(\large\color{black}{ \displaystyle \int_{f(a)}^{f(b)} {\rm G}\left(\color{red}{u}\right)du }\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4I meant \(du=f'(x)~dx\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4should i give ordinary, reg. example?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4\(\large\color{black}{ \displaystyle \int_{2}^{4} 2x\cdot (x^23)^{100}dx }\) you don't want to epand, do you? O~O you set \(u=x^23\) and then \(du=2x~dx\) (see how nice it is that you have that "2x" part in the integral) and when x=2, u=\((\color{red}{2}^23)=1\) (because u in relation to x is x²3) and when x=4, u=\((\color{red}{4}^23)=13\) So you get: \(\large\color{black}{ \displaystyle \int_{1}^{13} (u)^{100}du }\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4and then the rest in that example is obvious (note, that I change the limits of integration equivalently, so you don't need to sub back the x)

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0I did it in a different way..... (I dont understand why it should be wrong) dw:1440428321372:dw I multiplied the brackets

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1Please, wait....Let me digest it c:

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4dw:1440428430105:dw

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0dw:1440428440544:dw

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0why is \(u\) substitution dropped ?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4what do you mean ganeshie?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4dropped, as "dropped into the thread"  saying, why is it proposed? or, "dropper" as left out?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0im asking why are you expanding that out instead of simply using \(u\) substitution..

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4Well, arinda.. propsed that approach as it is also a posibility (in this case) for those that aren't familir with u sub. I just say that derivative of 3t^28t+3 by so I said u sub right off

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4just *see* (typos, oh my)

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0One question,

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4Abhisar, if you want another example let me know ...

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0Without u substitution... it can be solved this way right?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0Ahh okay nice :) its just that i felt this problem is begging for \(u\) substitution... expanding it out is somewhat pain...

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4Yes, arinda it is possible, certainly:)

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0okay.... :D

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4not that i would abide by that approach in such case.

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1One min, let me try this method on the original problem.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4Sure, go ahead:) \(!^\infty\)

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0I don't know about u substitution, could you guyz teach me?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4this is not your question, apologize to remind you;(

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4You can make your own tho' there are plenty people (just on this post, certainly on the site) that can help you much better than little me

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0ya @SolomonZelman but thanks to @Abhisar to raise this question, I got to know about u substitution!

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1Ok, so applying u substitution method on the original problem, I can choose any one as u. Correct?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4Well, you can choose anything technically, but I really don't promise it will be easier if you choose not the ideal one

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4You know that when you substitute u=f(x) in any case, then whatever the derivative of this f(x) is (we will reasonably call it f'(x)) then the new du replaces "f'(x) and dx"

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1ok, let's start with \(\sf 38t+3t^2\) as u, then \(du = 8+6t\).

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4Tell me what is the derivative (with respect to t) of \(3t^28t+3\) ?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4yes, correct LOL

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4and knowing that u=3t²8t+3 you can find the corresponding limits of integration when t=0, u=what? when t=4, u=what?

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0I am watching :D Nice.....

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1and knowing that u=3t²8t+3 you can find the corresponding limits of integration when t=0, u=3 when t=4, u=19

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4So, what you do then is: \(\large\color{black}{\displaystyle\int\limits_{0}^{4}\color{red}{(6t8)}(38t+3t^2)\color{red}{dt}}\) the red part is replaced by du and the `38t+3t²` piece is your you u, so you put that in too. Also, put in your new limits of integration that correspond to your "u". \(\large\color{black}{\displaystyle\int\limits_{3}^{19}(u)\color{red}{du}}\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4the `38t+3t²` is your \(\cancel{\rm you}\) u

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4did i confuse you by typing too much? (sorry for this habit then)

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1No it's actually helping me. Wait a sec.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4if you don't understand something, then don't hesitate to ask...

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1So I write, \(\large\color{black}{\displaystyle\int\limits_{3}^{19}\color{red}{(8+6t)}}(3t4t^2+t3)\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4no, not really... you mixed it up a bit.

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0its without 8+6t

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0the same expression you wrote except that!

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1\[\int\limits_{3}^{13}(3t²8t+3).du\]

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0right @SolomonZelman ? no 3 to 19 !! @Abhisar

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1\[\int\limits_{3}^{19}(3t²8t+3).du\] Sorry , Typo

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0yup, hope so

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4\(\large\color{black}{\displaystyle\int\limits_{0}^{4}(8+6t)(3t^28t+3)dt}\) You are starting from \(\color{blue}{u=3t^28t+3}\) (You are sort of doing a chain rule for u) \(\large\color{black}{\displaystyle\frac{du}{dt}=6t8}~~~~\Longrightarrow\) \(\large\color{black}{\displaystyle \color{blue}{du= (8+6t)dt}}\) \(\large\color{black}{\displaystyle x=0~~~\rightarrow~~~u=3(0)^28(0)+3~~~\rightarrow~~~u=3}\) \(\large\color{black}{\displaystyle x=0~~~\rightarrow~~~u=3(4)^28(4)+3~~~\rightarrow~~~u=19}\) Now you put in the following into the integal: 1) "u" instead of "3t²8t+3" 2) "du" instead of "(8+6t)•dt" 3) Lower limit of u=3 instead of "t=0" 4) Upper limit of u=19 instead of "t=4" YOU GET: \(\large\color{red}{\displaystyle\int\limits_{3}^{19}(u)du}\)

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0nice concept @SolomonZelman I totally got it! WOW!

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4Note that "8+6t" and "dx" are going away, being replaced by "du".

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1Ok, so what will be the final answer O.O

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4well what is the integral of u (or, u¹) ?

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0replace t by u, dt by du and put 3 and 19 for the limit

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4\(\LARGE \rm \color{black}{\displaystyle\int\limits_{3}^{19}(u)du=\left.\frac{u^{1\color{royalblue}{+1}}}{1\color{royalblue}{+1}}~\right}^{_{19}}_{^3}\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4arinda what you said is just make everything the same but instead of using "u", use "t" it is going to be the same exact thing, but there is no need in doing that

arindameducationusc
 one year ago
Best ResponseYou've already chosen the best response.0okay @SolomonZelman

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1oh.....ok, !!!!! u=176. so now \(\sf 3t^2−8t+3=176\) right?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4why are you doing that, Abhisar?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4oh the result for the integral is 176 you mean?

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1Ooops..my apologies, yeah...

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4yes, and just to clarify, we are NOT substituting back the t, once we found the equivalent limits of integration for u and placed THEM in.

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1ok, one final question. In what cases we can use this method?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4In many cases. I really want to through a lot of examples rather than giving a verbal definition.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4\(\large\color{black}{ \displaystyle \int{}\sin^{80}(x)\cos(x)~dx }\) (you can see that u=sin(x), and du=cos(x) dx) AND NOTE: when you find the integral of u, yuo DO need to sub back the x into the result. from now i will post just the type of problems, not their solutions (but if you will be unable to do one of them, then go ahead and ask in this or if you like  a different post) \(\large\color{black}{ \displaystyle \int{}8x^3\sqrt{x^4+5} ~dx }\) \(\large\color{black}{ \displaystyle \int{}x\cos(x^2)~dx}\) (I am using x, but it can be any other letter, I am just used to x) \(\large\color{black}{ \displaystyle \int{}x(x^41)~dx}\) (this is a little more advanced tho) want more?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4mostly, the once that have the derivative of u sitting in the integral.

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1Let me do them first, I think they will be enough. But thank you very much for the help. I really appreciate it, you have been tutoring me since an hour, that means a lot c:

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4You are always welcome!

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1I will post my answers in a new thread since this one is becoming annoying to handle c:

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4one note: By indefinite integrals, you always need to sub in the original variable back (after you integrate with the substituted variable).

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.4alright, if you like to use a new thread:) Whatever you like!

Abhisar
 one year ago
Best ResponseYou've already chosen the best response.1Yeah c; and thanks a bunch again c:
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