How do you evaluate the definite integral #int (dx/(xsqrtlnx))# from # [1,e^3]#?
1 Answer
Explanation:
We have the definite integral:
#int_1^(e^3)dx/(xsqrtlnx)#
This is a prime time to use substitution. Note that in the integrand, we have both
So, if we let
Since we are moving from an integral in terms of
Thus the bound of
This integral is about to change dramatically. We see that:
#int_1^(e^3)dx/(xsqrtlnx)=int_0^3(du)/sqrtu#
In order to integrate this, rewrite
#int_0^3(du)/sqrtu=int_0^3u^(-1/2)du=[u^(-1/2+1)/(-1/2+1)]_0^3=[u^(1/2)/(1/2)]_0^3=[2sqrtu]_0^3#
#=2sqrt3-2sqrt0=2sqrt3#