What is the integral of #int_1^edy/(y*sqrt(1+(lny)^2#? Evaluate the integrals using an appropriate substitution and then a trigonometric substitution.

#int_1^edy/(y*sqrt(1+(lny)^2#

1 Answer
Feb 8, 2018

#int_1^(e) dy/[y*sqrt(1+(Lny)^2)]=ln(sqrt2+1)#

Explanation:

#int_1^(e) dy/[y*sqrt(1+(Lny)^2)]#

After using #Lny=tanu# and #dy/y=(secu)^2*du# transforms, this integral became

#int_0^(pi/4) ((secu)^2*du)/sqrt(1+(tanu)^2)#

=#int_0^(pi/4) ((secu)^2*du)/sqrt((secu)^2)#

=#int_0^(pi/4) ((secu)^2*du)/secu#

=#int_0^(pi/4) secu*du#

=#int_0^(pi/4) (secu*(secu+tanu)*du)/(secu+tanu)#

=#[Ln(secu+tanu)]_0^(pi/4)#

=#Ln(sqrt2+1)#