Find the approximated integral value of an equation x^3-2x+1 using Simpson's 3/8 rule
a = 2 and b = 4
Step value (h) = 0.5

Solution:
Equation is `f(x)=x^3-2x+1`

`a=2`

`b=4`

The value of table for `x` and `f(x)`

`x`	`f(x)`
`x_0=2`	`f(x_(0))=f(2)=5`
`x_1=2.5`	`f(x_(1))=f(2.5)=11.625`
`x_2=3`	`f(x_(2))=f(3)=22`
`x_3=3.5`	`f(x_(3))=f(3.5)=36.875`
`x_4=4`	`f(x_(4))=f(4)=57`

Method-1:
Using Simpson's `3/8` Rule



`int f(x) dx=(3Delta x )/8 [f(x_(0))+3f(x_(1))+3f(x_(2))+2f(x_(3))+f(x_(4))]`

`f(x_(0))=5`

`3f(x_(1))=3*11.625=34.875`

`3f(x_(2))=3*22=66`

`2f(x_(3))=2*36.875=73.75`

`f(x_(4))=57`

`int f(x) dx=(3xx0.5)/8 *(5+34.875+66+73.75+57)`

`=(3xx0.5)/8 *(236.625)`

`=44.3672`

Solution by Simpson's `3/8` Rule is `44.3672`

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Method-2:
Using Simpson's `3/8` Rule



`int f(x) dx=(3Delta x )/8 [(f(x_(0))+f(x_(4)))+2(f(x_(3)))+3(f(x_(1))+f(x_(2)))]`

`=(3xx0.5)/8 [(5 +57)+2xx(36.875)+3xx(11.625+22)]`

`=(3xx0.5)/8 [(5 +57)+2xx(36.875)+3xx(33.625)]`

`=(3xx0.5)/8 [(62)+(73.75)+(100.875)]`

`=44.3672`

Solution by Simpson's `3/8` Rule is `44.3672`

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