Find the approximated integral value of an equation 1/x using Simpson's 1/3 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 Simpsons `1/3` Rule



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

`f(x_(0))=1`

`4f(x_(1))=4*0.8=3.2`

`2f(x_(2))=2*0.6667=1.3333`

`4f(x_(3))=4*0.5714=2.2857`

`f(x_(4))=0.5`

`int f(x) dx=0.25/3*(1+3.2+1.3333+2.2857+0.5)`

`=0.25/3*(8.319)`

`=0.6933`

Solution by Simpson's `1/3` Rule is `0.6933`

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Method-2:
Using Simpsons `1/3` Rule



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

`=0.25/3 [(1 +0.5)+4xx(0.8+0.5714)+2xx(0.6667)]`

`=0.25/3 [(1 +0.5)+4xx(1.3714)+2xx(0.6667)]`

`=0.25/3 [(1.5)+(5.4857)+(1.3333)]`

`=0.6933`

Solution by Simpson's `1/3` Rule is `0.6933`

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