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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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