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Math · Advanced Math
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Consider approximating integrals of the form

I(f)=int_{0}^{1}sqrt{x}f(x)dx

in which f(x) has several continuous derivatives on [0,1]

a) Find a formula I(f)=int_{0}^{1}sqrt{x}f(x)dx=w_1f(x_1)=I_1(f) which is exact if f(x) is any linear polynomial.

b) To find a fomula I(f)=int_{0}^{1}sqrt{x}f(x)dx=w_1f(x_1)+w_2f(x_2)=I_2(f) which is exact for all polynomials of degrees set up a system of four equations with unknowns w1, w2, x1 and x2.

c) Apply I_1 and I_2 to evaluate

re dx

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