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Determine coefficients $A_k ; k=1,2,3,4$ in the following quadratic formula $$\int_{-1}^{1}f(x)dx=A_1f(-1)+A_2f(1)+A_3f'(-1)+A_4f(1)+R$$ such that it has the maximum possible algebraic degree of accuracy. Evaluate an error, and using the obtained formula evaluate the following integral:

$$\int_0^{\pi/2}\sin xdx.$$

Could someone explain in details how to solve these types of problems?

user300045
  • 3,449
  • Substitute $f(x)$ for $1,x,x^2,x^3,x^4$. Put $R=0$ in all cases. You get a system of 4 linear equations in the four unknowns $A_1,A_2,A_3,A_4$. – OR. Jun 17 '17 at 16:11
  • @Mlazhinka Shung Gronzalez LeWy , Could you give a complete answer? – user300045 Jun 17 '17 at 16:17
  • Is the $A_4f(1)$ missing a prime? – OR. Jun 17 '17 at 16:21
  • @Mlazhinka Shung Gronzalez LeWy, No. – user300045 Jun 17 '17 at 16:22
  • Ok, then we might need to hope for less. Call $A_2f(1)+A_4f(1) = Bf(1)$. And for any solution for $B$ $A_2,A_4$ can take any values that add up to $B$. – OR. Jun 17 '17 at 16:25
  • @Mlazhinka Shung Gronzalez LeWy, Ok, could you give a complete answer now? – user300045 Jun 17 '17 at 16:32
  • yes, one sec. Grocery shopping first, if you haven't got an answer I will post one. – OR. Jun 17 '17 at 16:33
  • Ok. The problem is likely badly copied but well ... To solve it as if the quadrature were $A_1f(-1)+Bf(1)+A_3f'(-1)$ we need the equation to be satisfied for $f(x)=1,x,x^2,...,x^n$, with $R=0$ for $n$ as large as possible. Replace those $f$ and you get many equations. So, you need to take the first $n$ equations with $n$ as large as possible such that there are solutions. For that largest $n$, solve the system and find one solution for$A_1,B,A_3$. Finally Evaluate $A_1\sin(-1)+B\sin(1)+A_3\cos(-1)$. – OR. Jun 17 '17 at 19:08
  • @Mlazhinka Shung Gronzalez LeWy, Could you please give a full answer? – user300045 Jun 17 '17 at 20:26

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