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

Solution:
Equation is `f(x)=2x^3-4x+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)=9`
`x_1=2.5`	`f(x_(1))=f(2.5)=22.25`
`x_2=3`	`f(x_(2))=f(3)=43`
`x_3=3.5`	`f(x_(3))=f(3.5)=72.75`
`x_4=4`	`f(x_(4))=f(4)=113`

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))=9`

`3f(x_(1))=3*22.25=66.75`

`3f(x_(2))=3*43=129`

`2f(x_(3))=2*72.75=145.5`

`f(x_(4))=113`

`int f(x) dx=(3xx0.5)/8 *(9+66.75+129+145.5+113)`

`=(3xx0.5)/8 *(463.25)`

`=86.8594`

Solution by Simpson's `3/8` Rule is `86.8594`

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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 [(9 +113)+2xx(72.75)+3xx(22.25+43)]`

`=(3xx0.5)/8 [(9 +113)+2xx(72.75)+3xx(65.25)]`

`=(3xx0.5)/8 [(122)+(145.5)+(195.75)]`

`=86.8594`

Solution by Simpson's `3/8` Rule is `86.8594`

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