the curve of ylnx is defined on the interval 1a where a is a

Solution

y = ln(x) y\' = (1/x) L = Int (1 to a) [ sqrt(1 + (1/x)^2) ] dx L = Int (1 to a) [ sqrt(1 + (1/x)^2) ] dx = Int (1 to a) [ sqrt(1 + (1/x^2)) ] dx = Int (1 to a) [ sqrt( (x^2 + 1) / x^2) ) ] dx =Int (1 to a) [(1/x)*sqrt(x^2 + 1) ] dx You need a table of integrals... Int [(1/u)sqrt(b^2 + u^2)] du = sqrt(b^2 + u^2) - a*ln | [ab+ sqrt(b^2 + u^2)] / u] | + C In this integral,b= 1 and u = x Int (1 to a) [(1/x)*sqrt(x^2 + 1) ] dx = sqrt(1 + x^2) - ln | [1 + sqrt(1 + x^2)] / x] | (1 to a) = sqrt(1 + a^2) - ln | [1 + sqrt(1 + a^2)] / a] | - sqrt(2) - ln | [1 + sqrt(2)]] |