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# Compute the value of ∫ 1.4 0.2 ( sin x − I n x + e x ) Trapezoidal Rule (ii) Simpson’s (1/3)rd rule (iii) Simpson’s (3/8)th rule by dividing Into six subintervals. - Applied Mathematics 2

Compute the value of int _0.2^1.4 (sin  x - In x+e^x)  Trapezoidal Rule (ii) Simpson’s (1/3)rd rule (iii) Simpson’s (3/8)th rule by dividing Into six subintervals.

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#### Solution

let I = int _0.2 ^1.4 (sin x -"in"  x+e^x) dx

∴ n=6 ∴ h=(b-a)/n=(1.4-0.2)/6=1/5

 x_0=0.2 x_1=0.4 x_2=0.6 x_3=0.8 x_4=1.0 x_5=1.2 x_6=1.4 y_0=3.02 y_1=2.79 y_2=2.89 y_3=3.16 y_4=3.55 y_5=4.06 y_6=4.4

(i) Trapezoidal rule : I= h/2 [x+2R]        -----------------(1)

X = sum of extreme ordinates=7.42

R=sum of remaining ordinates = 16.45

I=1/5xx2 (7.42+2(16.45))              ……………….(from 1)

I=4.032

(ii) Simpson’s (1/3)^(rd) rule :

I=h/3[X+2E+40]                      ---------------(2)

X= sum of exterme ordinates= y_0+y_6=4.4+3.02=7.42

E= sum of even base ordinates = y_2+y_4=6.44

O=sum of odd base ordinates = y_1+y_3+y_5= 10.01

I=1/3xx5(7.42+2xx6.44+4xx10..01)

I = 4.022

(iii) Simpson’s (3/8)^(th) rule

I=3h/8[X+2T+3R]                            -------------(3)

X= sum of extreme ordinates=y_0+y_6=4.4+3.02=7.42

T= sum of multiple of three base ordinates=y_3=3.16

R= sum of remaining ordinates=y_1+y_2+y_4+y_5=13.49

∴ I=(3xx1)/(8xx5) [7.42+2xx3.16+3xx13.49]

[I=4.02075]

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
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