How do you find the definite integral of #(dx)/(xsqrt(ln(x))# from #[e^4, e^16]#?

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Answer 1 · 2018-05-07T14:15:36

#int_(e^4)^(e^16) dx/(xsqrt(lnx)) = 4 #

Substitute #t = lnx#, #dt = dx/x#.

The limits of integration become:

#lim_(x->e^4) t(x) = 4#

#lim_(x->e^16) t(x) = 16#

so:

#int_(e^4)^(e^16) dx/(xsqrt(lnx)) = int_4^16 dt/sqrtt#

#int_(e^4)^(e^16) dx/(xsqrt(lnx)) =[2sqrtt]_4^16 #

#int_(e^4)^(e^16) dx/(xsqrt(lnx)) =8-4 = 4 #