Question

Let x In I dx. a) Find the exact value of 1 b) Use composite trapezoidal rule with n = 4 subintervals to approximatel. Calculate the exact error c) Use composite simpson's rule with n = 4 subintervals to approximatel. Calculate the d) Use composite simpson's rule with n = 6 subintervals to approximate I. Calculate the exact error exact error

Answer 1

2 I - relnr dre L fost finding exact value of I we use use inicgreetion by parts on fa n.enn da. det u = en e and vi =dv u's du V dr V 22 2 By using formule Suva UV suv. Substituting above Veilues in the fosimula. Suvi al UV -suv nilane - s t t die »?. lnic Ž freede 2 2 I x?. enn re² 4 tc C 2 Hence exact value of I is 1 re? Inne

2. 2 با ہوا۔ (Ince ] 2 2) n (2) 2 + e-usen() - ( (| 13862 - | - | ( (. O.1362 exact value of I is 362 لون ا

© 2 ſ S a envedre Trapezoidal rule with n=4 Now using trapezoidal rule to estimate I've Inn die using 4 intervals S The endpoints of the subintenvals consist of elements of the set {moro, Ne, r3 na} - {tik :2} S 2 and oxa b-a 2-1 S n 1/4 4 Thus Tha 2 re.innela * Ime day solna.dn = I Over [f(x) + 2 f(x, )4 2f (42)+.. f (xn=;) + f(un) ] (964) f(t) + 2 + ( +2+ (3) 2+(2) + f (2) (3) + 4.2 ]

T4 11 2 O + 05578 + 1: 2163 + 19586+.1. 3862 g [ ТА - 0.639900 Now eu107 Ersor Testinnutel vake Tcalculated value Teakulated value 0:1362 0.6399 0:1362 3.6982.

2 (C) T S x.lna da Now using simpsons rule to estimate S² re enn de using 4 intervals. The Simpson's rule states that Дх S. * $(re)dre 3 [ f() +21 f(*, D + 2 f(72) + 4f(x) + 2F (MA) + flun n where On b-a n IJO have a-1, b=2, na 4 so sn = 2-) - 24 4 Divide the interval [1,2] into * = 4 subintervals of length on a with the fol following IN endpoints' 1 a - L 1, 2:6 7 4 2 2 Now we just evaluate the function at these endpoints :-

(")) } f(1) = TL Sirr 4f (11.) = .4f€): son af (12) = 21 2x3 In (2) - 1.2.639 2 (au)th hf (1) 7 In (30) = 3.9173 ("x)) F (2) 2 An (2) = 1.3862 finally just sum up the above values and multiply by An ut E uk Sirene det ti[ + of t. 2157 +1.2163+ 3.1737 1.3862 ا = 0.63630 Now Eroor Calutted value Estimated Value Calculated value 0.1362 0.6363 ZOET.O Error Ć 3.6718

d ) S. alreada have ste, b: 2, ng An 2-1 6 iuto n=6 divide the Intervals [1, 2] Subintervals of length in following end points . with the - 7 6 4 2=b ; زه وار Just evaluate the function at these endpoint Fino) f(1) =0 14. in (3) 507193 Afinu) 46(2) 2 + (11) = 28(4): in (): 0.7671 if(n) af ( 2 ) = 6in G = 2.4327 2f (14) = f (5/3) = 10 In 10 In () 4f (Ms) = af 1%) = ??t (0) = 4.4949 Fano) = f(2) - 2 In (2) = 1.3862 17027

finally : Just sumup the above values and multiply by ДИ 3 16 gir brinda [ot 0+ 0.7193 + 0.76 7.1 +2:4327 + 18 1.7027 + 2.69449 + 1.346.2 0.6362 Now Error Calculated Value Estimated valuo Caleeleped value 0:1362 0.6362 ..252 Froor . 3.67.10