Doubt from Definite Integration

No idea on where to start and what to do…
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Tried using both trapzoidal and simson method but getting integral in terms of f(-1), f(0) and f(1) from there. Not sure from where these ±1/√3 is coming. :thinking::thinking:

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take f(x) = Ax^3 + Bx^2 + Cx + D now \int_{-1}^{1} f(x)\ dx = \frac{2B}{3} + 2D which if you see is equal to f(\frac{-1}{\sqrt3}) + f(\frac{1}{\sqrt3})

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By putting values I am also getting. I was wondering if there’s a theoritical way to get to the result.

Or is observing is the only way as you said?

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yeh theoretical hi toh h :thinking:

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they have mentioned no error so… i think it’s the only way out

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I meant ye toh apne ne observe karke find kara na ki f(-1/√3) + f(1/√3) ke equal aa raha hh.

Like this is the same problem and here we are required to “show” ki ye values aayegi
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i think that is the only way still, they are limiting to x^3 because of this

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Iss topic ke liye koi video source pata hh kya aapko @Ldrago? I searched but mila nahi.

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bro tum series ka expansion dhoond rahe ho kya, maclaren series se sinx , e^x , tan-1x , etc derive kiye hai maine , for series at right of the page u have to equate standard sinx , cosx expansion ko euler series ( series of their roots ) se equate karna hai Proof 1/1+1/4+1/9+1/16+1/25+...=(pi^2)/6 - YouTube also i landed up on 3bib at the time i was studying limits first Why is pi here? And why is it squared? A geometric answer to the Basel problem - YouTube see this only if u hv time

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bro sabka proof pata kr lo or try kro prove krne ka
Sandwich use krke or integration or Expansion
iss sab se aa jata h
yaad nhi krne h ye except 1 or 2

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https://www.quora.com/What-is-displaystyle-int_-0-1-frac-ln-1-x-x-2-ldots-x-n-x-mathrm-dx/answer/Hallie-Black-4?ch=3&share=c6315741&srid=hC5lU
Here is a problem which uses one of these summations.

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Second summation is in GMP… subjective problem using sandwich

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Nahi ye nahi. Ye sab kar rakha hh maine.

I got a little confused ki ye koi naya topic hh. But jab question solve kiye toh pata chala ki isme sirf vo π²/6 and ln 2 wali series yaad rakhni hh mujhe.

And iss topic ka conclusion ye nikla ki kisi bhi infinite power series ko integrate kar sakte as long as it is converging in the given interval.

And 1+1/2² + 1/3² + … uska proof dekh liya maine ab. Thanks @PRANAV_MANIYAR @Deadpool @VictoryGod

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Please try this one also.

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From given information , you can form 3 simultaneous equation in term of three variables (A,B,C)
Value of Determinant made by coefficients is non - zero for non zero value of h.
So it will always have a solution

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Thanks got it. @Devansh_Jain :fire:

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I don’t think this qualifies as a doubt but still…

What does he mean by “independence of choice of points Ei” (in last third line)?

Rest things are understandable except this phrase which I can’t comprehend and he has used is at many places.

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He basically defined reimann sums…some sort of limit to the vicinity is \mathsf{Ei}

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Reimann sums is basically limit of sum right?

Do I ignore this phrase and move forward? This wouldn’t affect any question I believe.

Good thanks @Sherlock_Holmes

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But Ye JEE ke syllabus me nahi hai mere Bhaiya ne bola udhar first year me padhaya Jaata Hai.

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