Question

alpha = 3

beta =2 Can you solve it in a hour please Thank you very much.

3. (15p.) Approximate the following integral using the two-point Gaussian quadrature rule (x +a)e(2-1)2-Bdx 2 0

Answer 1

The answer is given as

Soid we have to approximate ( the following integral using the two-point braussian quadrature babe The integral is } (x+4)² e 17-18-td d=3 & ß =2 ß =2 hence Since

Integral becomes 3² (x+372 e (x-1) ²-2 dx let 5 (x+3y2 e(x-1²-2 e = dx & let f(x) (2-102_2 (x+32 e traussion Now The two-point quadrature rule f for sofia) da ist goflade e, f(xi) & Caf (42) where ci- b-a & 2 (=b-a 2 x b-a 2 + bta 2 Az N ba (it) + bta 2 Now you our case, we have f(x) (x+372 (x-1²-2

and a =0;b=2 hence 디는 자 을 21 디 디 ㄸ디 끊) + (+) 게 . ㅗ + 형 지 - u - - 057735 지 =42265 and 2 1 (ㄶ ) + f 0-57735 =157735

f(1) = f(0.422 65) (0.42265-10²2 (0.42265 +3)2 e = (3.42265)?e60.57735)-2 = (3.422653² & 1.66666 -1.66666 = 11.71453 e = 11:71453 (0.18.887) f(32) 2.21252 and f(az) = f (1.57735) - (1.57735 +3) 641.57735-132-2 = (4.5773530-33333).-2 = 20.95213 ( ē 1.66667) 20.95213 (0.18867)

f(22) = 3.95722 Grauss hence from two-point quadrature lule - 5 (x+3) 13 dx š ci fra) + (2 fron) ♡! (2.21252) +1(3.95722) 4. (2.21252 +3-95722) ñ 6.16974 hence, the of approximate Value tegral is - the given in | 5 ore (4+3)? €2-13? 2 (x-1)-2 da s 6.16974