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`y=lnx , [1,5]` Find the arc length of the curve over the given interval.

 Arc length of curve can be denoted as "`S` ". We can determine it by using integral formula on a closed interval [a,b] as: `S = int_a^b ds`


where:


`ds = sqrt(1+ ((dy)/(dx))^2 )dx`  ` if y=f(x)`


or


`ds = sqrt(1+((dx)/(dy))^2) dy if x=h(y)`


`a` = lower boundary of the closed interval


`b` =upper boundary of the closed interval



From the given problem: `y =ln(x), [1,5]` , we determine that the boundary values are:


`a= 1` and `b=5`


Note that `y= ln(x)` follows `y=f(x)` then the formula we will follow can be expressed as `S =int_a^bsqrt(1+ ((dy)/(dx))^2 )dx`


For the derivative of ` y` or `(dy)/(dx)` , we apply the derivative formula for logarithm:


`d/(dx)y= d/(dx) ln(x)`


`(dy)/(dx)= 1/x`


 Then` ((dy)/(dx))^2= (1/x)^2`  or `1/x^2` .


Plug-in the values  on integral formula for arc length of a curve, we get:


`S =int_1^5sqrt(1+1/x^2 )dx`


Let `1 = x^2/x^2` then we get:


`S=int_1^5sqrt(x^2/x+1/x^2 )dx`


    `=int_1^5sqrt((x^2+1)/x^2 )dx`


    `=int_1^5sqrt(x^2+1)/sqrt(x^2 )dx`


    `=int_1^5sqrt(x^2+1)/sqrt(x^2 )dx`


    `=int_1^5sqrt(x^2+1)/xdx`


From the integration table,  we follow the formula for rational function with roots:


`int sqrt(x^2+a^2)/x dx = sqrt(x^2+a^2)-a*ln|(a+sqrt(x^2+a^2))/x|` .


Applying the integral formula with a^2=1 then a=1, we get:


`int_1^5sqrt(x^2+1)/xdx = [sqrt(x^2+1)-1*ln|(1+sqrt(x^2+1))/x|]|_1^5`


                     `= [sqrt(x^2+1)-ln|(1+sqrt(x^2+1))/x|]|_1^5`


Apply the definite integral formula: `F(x)|_a^b= F(b)-F(a)` .


`[sqrt(x^2+1)-ln|(1+sqrt(x^2+1))/x|]|_1^5`


`=[sqrt(5^2+1)-ln|(1+sqrt(5^2+1))/5|]-[sqrt(1^2+1)-ln|(1+sqrt(1^2+1))/1|]`


`=[sqrt(25+1)-ln|(1+sqrt(25+1))/5|]-[sqrt(1+1)-ln|(1+sqrt(1+1))/1|]`


`=[sqrt(26)-ln|(1+sqrt(26))/5|]-[sqrt(2)-ln|1+sqrt(2)|]`


`=sqrt(26)-ln|(1+sqrt(26))/5| -sqrt(2)+ln|1+sqrt(2)|`


Apply logarithm property: `ln(x)-ln(y) = ln(x/y)` .


`S =sqrt(26)-sqrt(2)+ln|1+sqrt(2)|-ln|(1+sqrt(26))/5|`


`S =sqrt(26)-sqrt(2)+ln|(1+sqrt(2))/(((1+sqrt(26))/5))|`


`S =sqrt(26)-sqrt(2)+ln|(5*(1+sqrt(2)))/(1+sqrt(26))|`


`S =sqrt(26)-sqrt(2)+ln|(5+5sqrt(2))/(1+sqrt(26))|` 


`S~~4.37`

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