June 10th, 2017, 06:20 AM  #1 
Newbie Joined: Jun 2017 From: France Posts: 3 Thanks: 0  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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