Find the approximated integral value of an equation 1/x using Simpson's 3/8 rule
a = 1 and b = 2
Step value (h) = 0.25

Solution:
Equation is `f(x)=(1)/(x)`

`a=1`

`b=2`

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

`x`	`f(x)`
`x_0=1`	`f(x_(0))=f(1)=1`
`x_1=1.25`	`f(x_(1))=f(1.25)=0.8`
`x_2=1.5`	`f(x_(2))=f(1.5)=0.6667`
`x_3=1.75`	`f(x_(3))=f(1.75)=0.5714`
`x_4=2`	`f(x_(4))=f(2)=0.5`

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))=1`

`3f(x_(1))=3*0.8=2.4`

`3f(x_(2))=3*0.6667=2`

`2f(x_(3))=2*0.5714=1.1429`

`f(x_(4))=0.5`

`int f(x) dx=(3xx0.25)/8 *(1+2.4+2+1.1429+0.5)`

`=(3xx0.25)/8 *(7.0429)`

`=0.6603`

Solution by Simpson's `3/8` Rule is `0.6603`

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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)))]`

`=(3xx0.25)/8 [(1 +0.5)+2xx(0.5714)+3xx(0.8+0.6667)]`

`=(3xx0.25)/8 [(1 +0.5)+2xx(0.5714)+3xx(1.4667)]`

`=(3xx0.25)/8 [(1.5)+(1.1429)+(4.4)]`

`=0.6603`

Solution by Simpson's `3/8` Rule is `0.6603`

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