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June 10th, 2017, 06:20 AM   #1
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Digital integration

Good evening to all of you,I'm French and I answered the first question of the exo below but I would have wanted me to explain how to do for question 2 if possible.
Here is this exo:
Let the function $\displaystyle f(x) = \frac{1}{1 + x ^ 2}$

1) Calculate $\displaystyle I =\int _{- 1}^1 f(x)dx $
2) Construct the following formula:

$\displaystyle I=\int_{-1}^1 f (x) dx \approx J_1(x) =a.f(-1/2)+b.f(0)+c.f(1/2)$

Precision 2 relationship.
Show that its accuracy goes up to order 3.

There are other issues, but it's the 2 one that interests me at the moment.
I tried to answer through a similar exo seen going on.

1) For this question, the integral is $\displaystyle \tan^{- 1}(x) $ and we have $\displaystyle \tan^{- 1}(1)-\tan^{- 1}(-1)$ = $\displaystyle \frac {\ pi}{2}$ therefore 90 degrees.

Then for question 2) I said that since the precision goes to the order 3, then we can say that the polynomials of order 3 are for canonical form {1, x, x ^ 2, x ^ 3}.

Then we put at the beginning f (x) = 1 which implies that $\displaystyle \int _ {- 1} ^ 1 dx $ = 2 of or $\displaystyle a.f(-1/2) .1 + bf (0) .1 + cf (1/2) .1 = 2$

Then we put f (x) = x so ...
...
Finally f (x) = $\displaystyle x^3$
Someone could help me please?

Last edited by skipjack; June 10th, 2017 at 03:22 PM.
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